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Every planar graph not containing a triangle is 3-colourable. | Null | No |
Proposition 5.2.1. Every graph \( G \) with \( m \) edges satisfies\n\n\[ \chi \left( G\right) \leq \frac{1}{2} + \sqrt{{2m} + \frac{1}{4}}. \] | Proof. Let \( c \) be a vertex colouring of \( G \) with \( k = \chi \left( G\right) \) colours. Then \( G \) has at least one edge between any two colour classes: if not, we could have used the same colour for both classes. Thus, \( m \geq \frac{1}{2}k\left( {k - 1}\right) \) . Solving this inequality for \( k \), we obtain the assertion claimed. | No |
Proposition 5.2.2. Every graph \( G \) satisfies\n\n\[ \chi \left( G\right) \leq \operatorname{col}\left( G\right) = \max \{ \delta \left( H\right) \mid H \subseteq G\} + 1. \] | Null | No |
Let \( G \) be a graph and \( k \in \mathbb{N} \) . Then \( \chi \left( G\right) \geq k \) if and only if \( G \) has a \( k \) -constructible subgraph. | Proof. Let \( G \) be a graph with \( \chi \left( G\right) \geq k \) ; we show that \( G \) has a \( k \) - constructible subgraph. Suppose not; then \( k \geq 3 \) . Adding some edges if necessary, let us make \( G \) edge-maximal with the property that none of its subgraphs is \( k \) -constructible. Now \( G \) is not a complete \( r \) -partite graph for any \( r \) : for then \( \chi \left( G\right) \geq k \) would imply \( r \geq k \), and \( G \) would contain the \( k \) -constructible graph \( {K}^{k} \) .\n\nSince \( G \) is not a complete multipartite graph, non-adjacency is not an equivalence relation on \( V\left( G\right) \) . So there are vertices \( {y}_{1}, x,{y}_{2} \) such that\n\n\( x,{y}_{1},{y}_{2} \) \( {y}_{1}x, x{y}_{2} \notin E\left( G\right) \) but \( {y}_{1}{y}_{2} \in E\left( G\right) \) . Since \( G \) is edge-maximal without a \( k \) -constructible subgraph, each edge \( x{y}_{i} \) lies in some \( k \) -constructible\n\n\( {H}_{1},{H}_{2} \) subgraph \( {H}_{i} \) of \( G + x{y}_{i}\left( {i = 1,2}\right) \) .\n\n\( {H}_{2}^{\prime } \) Let \( {H}_{2}^{\prime } \) be an isomorphic copy of \( {H}_{2} \) that contains \( x \) and \( {H}_{2} - {H}_{1} \) \( {v}^{\prime } \) etc. but is otherwise disjoint from \( G \), together with an isomorphism \( v \mapsto {v}^{\prime } \) from \( {H}_{2} \) to \( {H}_{2}^{\prime } \) that fixes \( {H}_{2} \cap {H}_{2}^{\prime } \) pointwise. Then \( {H}_{1} \cap {H}_{2}^{\prime } = \{ x\} \), so\n\n\[ H \mathrel{\text{:=}} \left( {{H}_{1} \cup {H}_{2}^{\prime }}\right) - x{y}_{1} - x{y}_{2}^{\prime } + {y}_{1}{y}_{2}^{\prime } \]\n\nis \( k \) -constructible by (iii). One vertex at a time, let us identify in \( H \) each vertex \( {v}^{\prime } \in {H}_{2}^{\prime } - G \) with its partner \( v \) ; since \( v{v}^{\prime } \) is never an edge of \( H \) , each of these identifications amounts to a construction step of type (ii). Eventually, we obtain the graph\n\n\[ \left( {{H}_{1} \cup {H}_{2}}\right) - x{y}_{1} - x{y}_{2} + {y}_{1}{y}_{2} \subseteq G \]\n\nthis is the desired \( k \) -constructible subgraph of \( G \) . | Yes |
Every bipartite graph \( G \) satisfies \( {\chi }^{\prime }\left( G\right) = \Delta \left( G\right) \) . | Proof. We apply induction on \( \parallel G\parallel \) . For \( \parallel G\parallel = 0 \) the assertion holds.\n\nNow assume that \( \parallel G\parallel \geq 1 \), and that the assertion holds for graphs with fewer edges. Let \( \Delta \mathrel{\text{:=}} \Delta \left( G\right) \), pick an edge \( {xy} \in G \), and choose a \( \Delta \) -\n\n\( \Delta ,{xy} \)\n\nedge-colouring of \( G - {xy} \) by the induction hypothesis. Let us refer to the edges coloured \( \alpha \) as \( \alpha \) -edges, etc. \( \alpha \) -edge\n\nIn \( G - {xy} \), each of \( x \) and \( y \) is incident with at most \( \Delta - 1 \) edges. Hence there are \( \alpha ,\beta \in \{ 1,\ldots ,\Delta \} \) such that \( x \) is not incident with an \( \alpha ,\beta \) \( \alpha \) -edge and \( y \) is not incident with a \( \beta \) -edge. If \( \alpha = \beta \), we can colour the edge \( {xy} \) with this colour and are done; so we may assume that \( \alpha \neq \beta \) , and that \( x \) is incident with a \( \beta \) -edge.\n\nLet us extend this edge to a maximal walk \( W \) from \( x \) whose edges are coloured \( \beta \) and \( \alpha \) alternately. Since no such walk contains a vertex twice (why not?), \( W \) exists and is a path. Moreover, \( W \) does not contain \( y \) : if it did, it would end in \( y \) on an \( \alpha \) -edge (by the choice of \( \beta \) ) and thus have even length, so \( W + {xy} \) would be an odd cycle in \( G \) (cf. Proposition 1.6.1). We now recolour all the edges on \( W \), swapping \( \alpha \) with \( \beta \) . By the choice of \( \alpha \) and the maximality of \( W \), adjacent edges of \( G - {xy} \) are still coloured differently. We have thus found a \( \Delta \) -edge-colouring of \( G - {xy} \) in which neither \( x \) nor \( y \) is incident with a \( \beta \) -edge. Colouring \( {xy} \) with \( \beta \) , we extend this colouring to a \( \Delta \) -edge-colouring of \( G \) . | Yes |
There exists a function \( f : \mathbb{N} \rightarrow \mathbb{N} \) such that, given any integer \( k \), all graphs \( G \) with average degree \( d\left( G\right) \geq f\left( k\right) \) satisfy \( \operatorname{ch}\left( G\right) \geq k \) . | The proof of Theorem 5.4.1 uses probabilistic methods as introduced in Chapter 11. | No |
Lemma 5.4.3. Let \( H \) be a graph and \( {\left( {S}_{v}\right) }_{v \in V\left( H\right) } \) a family of lists. If \( H \) has an orientation \( D \) with \( {d}^{ + }\left( v\right) < \left| {S}_{v}\right| \) for every \( v \), and such that every induced subgraph of \( D \) has a kernel, then \( H \) can be coloured from the lists \( {S}_{v} \) . | Proof. We apply induction on \( \left| H\right| \) . For \( \left| H\right| = 0 \) we take the empty colouring. For the induction step, let \( \left| H\right| > 0 \) . Let \( \alpha \) be a colour occurring in one of the lists \( {S}_{v} \), and let \( D \) be an orientation of \( H \) as stated. The vertices \( v \) with \( \alpha \in {S}_{v} \) span a non-empty subgraph \( {D}^{\prime } \) in \( D \) ; by assumption, \( {D}^{\prime } \) has a kernel \( U \neq \varnothing \) . Let us colour the vertices in \( U \) with \( \alpha \), and remove \( \alpha \) from the lists of all the other vertices of \( {D}^{\prime } \) . Since each of those vertices sends an edge to \( U \), the modified lists \( {S}_{v}^{\prime } \) for \( v \in D - U \) again satisfy the condition \( {d}^{ + }\left( v\right) < \left| {S}_{v}^{\prime }\right| \) in \( D - U \) . Since \( D - U \) is an orientation of \( H - U \), we can thus colour \( H - U \) from those lists by the induction hypothesis. As none of these lists contains \( \alpha \), this extends our colouring \( U \rightarrow \{ \alpha \} \) to the desired list colouring of \( H \) . | Yes |
Every bipartite graph \( G \) satisfies \( {\operatorname{ch}}^{\prime }\left( G\right) = {\chi }^{\prime }\left( G\right) \) . | Proof. Let \( G = : \left( {X \cup Y, E}\right) \), where \( \{ X, Y\} \) is a vertex bipartition of \( G \) .\n\n(2.1.4)\n\nLet us say that two edges of \( G \) meet in \( X \) if they share an end in \( X \), and \( X, Y, E \) correspondingly for \( Y \) . Let \( {\chi }^{\prime }\left( G\right) = : k \), and let \( c \) be a \( k \) -edge-colouring of \( G \) .\n\nClearly, \( {\operatorname{ch}}^{\prime }\left( G\right) \geq k \) ; we prove that \( {\operatorname{ch}}^{\prime }\left( G\right) \leq k \) . Our plan is to use Lemma 5.4.3 to show that the line graph \( H \) of \( G \) is \( k \) -choosable. To apply the lemma, it suffices to find an orientation \( D \) of \( H \) with \( {d}^{ + }\left( e\right) < k \) for every vertex \( e \) of \( H \), and such that every induced subgraph of \( D \) has a kernel. To define \( D \), consider adjacent \( e,{e}^{\prime } \in E \), say with \( c\left( e\right) < c\left( {e}^{\prime }\right) \) . If \( e \) and \( {e}^{\prime } \) meet in \( X \), we orient the edge \( e{e}^{\prime } \in H \) from \( {e}^{\prime } \) towards \( e \) ; if \( e \) and \( {e}^{\prime } \) meet in \( Y \), we orient it from \( e \) to \( {e}^{\prime } \) (Fig 5.4.3).\n\n\n\nFig. 5.4.3. Orienting the line graph of \( G \)\n\nLet us compute \( {d}^{ + }\left( e\right) \) for given \( e \in E = V\left( D\right) \) . If \( c\left( e\right) = i \), say, then every \( {e}^{\prime } \in {N}^{ + }\left( e\right) \) meeting \( e \) in \( X \) has its colour in \( \{ 1,\ldots, i - 1\} \) , and every \( {e}^{\prime } \in {N}^{ + }\left( e\right) \) meeting \( e \) in \( Y \) has its colour in \( \{ i + 1,\ldots, k\} \) . As any two neighbours \( {e}^{\prime } \) of \( e \) meeting \( e \) either both in \( X \) or both in \( Y \) are themselves adjacent and hence coloured differently, this implies \( {d}^{ + }\left( e\right) < k \) as desired.\n\nIt remains to show that every induced subgraph \( {D}^{\prime } \) of \( D \) has a kernel. This, however, is immediate by the stable marriage theorem (2.1.4) for \( G \) , if we interpret the directions in \( D \) as expressing preference. Indeed, given a vertex \( v \in X \cup Y \) and edges \( e,{e}^{\prime } \in V\left( {D}^{\prime }\right) \) at \( v \), write \( e{ < }_{v}{e}^{\prime } \) if the edge \( e{e}^{\prime } \) of \( H \) is directed from \( e \) to \( {e}^{\prime } \) in \( D \) . Then any stable matching in the graph \( \left( {X \cup Y, V\left( {D}^{\prime }\right) }\right) \) for this set of preferences is a kernel in \( {D}^{\prime } \).\n\n(5.3.1)\n\nBy Proposition 5.3.1, we now know the exact list-chromatic index of bipartite graphs:\n\nCorollary 5.4.5. Every bipartite graph \( G \) satisfies \( {\operatorname{ch}}^{\prime }\left( G\right) = \Delta \left( G\right) \). | Yes |
Corollary 5.4.5. Every bipartite graph \( G \) satisfies \( {\operatorname{ch}}^{\prime }\left( G\right) = \Delta \left( G\right) \) . | Null | No |
Proposition 5.5.1. A graph is chordal if and only if it can be constructed recursively by pasting along complete subgraphs, starting from complete graphs. | Proof. If \( G \) is obtained from two chordal graphs \( {G}_{1},{G}_{2} \) by pasting them together along a complete subgraph, then \( G \) is clearly again chordal: any induced cycle in \( G \) lies in either \( {G}_{1} \) or \( {G}_{2} \), and is hence a triangle by assumption. Since complete graphs are chordal, this proves that all graphs constructible as stated are chordal.\n\nConversely, let \( G \) be a chordal graph. We show by induction on \( \left| G\right| \) that \( G \) can be constructed as described. This is trivial if \( G \) is complete. We therefore assume that \( G \) is not complete, in particular that \( \left| G\right| > 1 \) , and that all smaller chordal graphs are constructible as stated. Let \( a, b \in \) \( a, b \) \( G \) be two non-adjacent vertices, and let \( X \subseteq V\left( G\right) \smallsetminus \{ a, b\} \) be a minimal \( a - b \) separator. Let \( C \) denote the component of \( G - X \) containing \( a \), and put \( {G}_{1} \mathrel{\text{:=}} G\left\lbrack {V\left( C\right) \cup X}\right\rbrack \) and \( {G}_{2} \mathrel{\text{:=}} G - C \) . Then \( G \) arises from \( {G}_{1} \) and \( {G}_{1},{G}_{2} \) \( {G}_{2} \) by pasting these graphs together along \( S \mathrel{\text{:=}} G\left\lbrack X\right\rbrack \) .\n\nSince \( {G}_{1} \) and \( {G}_{2} \) are both chordal (being induced subgraphs of \( G \) ) and hence constructible by induction, it suffices to show that \( S \) is complete. Suppose, then, that \( s, t \in S \) are non-adjacent. By the minimality of \( X = V\left( S\right) \) as an \( a - b \) separator, both \( s \) and \( t \) have a neighbour in \( C \) . Hence, there is an \( X \) -path from \( s \) to \( t \) in \( {G}_{1} \) ; we let \( {P}_{1} \) be a shortest such path. Analogously, \( {G}_{2} \) contains a shortest \( X \) -path \( {P}_{2} \) from \( s \) to \( t \) . But then \( {P}_{1} \cup {P}_{2} \) is a chordless cycle of length \( \geq 4 \) (Fig. 5.5.1), contradicting our assumption that \( G \) is chordal. | Yes |
Theorem 5.5.3. (Chudnovsky, Robertson, Seymour & Thomas 2002) A graph \( G \) is perfect if and only if neither \( G \) nor \( \bar{G} \) contains an odd cycle of length at least 5 as an induced subgraph. | Null | No |
A graph is perfect if and only if its complement is perfect. | Applying induction on \( \left| G\right| \), we show that the complement \( \bar{G} \) of any perfect graph \( G = \left( {V, E}\right) \) is again perfect. For \( \mathcal{K} \) \( \left| G\right| = 1 \) this is trivial, so let \( \left| G\right| \geq 2 \) for the induction step. Let \( \mathcal{K} \) denote \( \alpha \) the set of all vertex sets of complete subgraphs of \( G \) . Put \( \alpha \left( G\right) = : \alpha \) , \( \mathcal{A} \) and let \( \mathcal{A} \) be the set of all independent vertex sets \( A \) in \( G \) with \( \left| A\right| = \alpha \) | No |
Lemma 5.5.5. Any graph obtained from a perfect graph by expanding a vertex is again perfect. | Proof. We use induction on the order of the perfect graph considered. Expanding the vertex of \( {K}^{1} \) yields \( {K}^{2} \), which is perfect. For the induction step, let \( G \) be a non-trivial perfect graph, and let \( {G}^{\prime } \) be obtained from \( G \) by expanding a vertex \( x \in G \) to an edge \( x{x}^{\prime } \) . For our proof that \( {G}^{\prime } \) is perfect it suffices to show \( \chi \left( {G}^{\prime }\right) \leq \omega \left( {G}^{\prime }\right) \) : every proper induced subgraph \( H \) of \( {G}^{\prime } \) is either isomorphic to an induced subgraph of \( G \) or obtained from a proper induced subgraph of \( G \) by expanding \( x \) ; in either case, \( H \) is perfect by assumption and the induction hypothesis, and can hence be coloured with \( \omega \left( H\right) \) colours.\n\nLet \( \omega \left( G\right) = : \omega \) ; then \( \omega \left( {G}^{\prime }\right) \in \{ \omega ,\omega + 1\} \) . If \( \omega \left( {G}^{\prime }\right) = \omega + 1 \), then\n\n\[ \chi \left( {G}^{\prime }\right) \leq \chi \left( G\right) + 1 = \omega + 1 = \omega \left( {G}^{\prime }\right) \]\n\nand we are done. So let us assume that \( \omega \left( {G}^{\prime }\right) = \omega \) . Then \( x \) lies in no \( {K}^{\omega } \subseteq G \) : together with \( {x}^{\prime } \), this would yield a \( {K}^{\omega + 1} \) in \( {G}^{\prime } \) . Let us colour \( G \) with \( \omega \) colours. Since every \( {K}^{\omega } \subseteq G \) meets the colour class \( X \) of \( x \) but not \( x \) itself, the graph \( H \mathrel{\text{:=}} G - \left( {X\smallsetminus \{ x\} }\right) \) has clique number \( \omega \left( H\right) < \omega \) (Fig. 5.5.2). Since \( G \) is perfect, we may thus colour \( H \) with \( \omega - 1 \) colours. Now \( X \) is independent, so the set \( \left( {X\smallsetminus \{ x\} }\right) \cup \left\{ {x}^{\prime }\right\} = V\left( {{G}^{\prime } - H}\right) \) is also independent. We can therefore extend our \( \left( {\omega - 1}\right) \) -colouring of \( H \) to an \( \omega \) -colouring of \( {G}^{\prime } \), showing that \( \chi \left( {G}^{\prime }\right) \leq \omega = \omega \left( {G}^{\prime }\right) \) as desired. | Yes |
Proposition 6.1.1. If \( f \) is a circulation, then \( f\left( {X,\bar{X}}\right) = 0 \) for every set \( X \subseteq V \) . | Proof. \( f\left( {X,\bar{X}}\right) = f\left( {X, V}\right) - f\left( {X, X}\right) = 0 - 0 = 0 \) . | Yes |
Corollary 6.1.2. If \( f \) is a circulation and \( e = {xy} \) is a bridge in \( G \), then \( f\left( {e, x, y}\right) = 0 \) . | Null | No |
Proposition 6.2.1. Every cut \( \left( {S,\bar{S}}\right) \) in \( N \) satisfies \( f\left( {S,\bar{S}}\right) = f\left( {s, V}\right) \) . | Proof. As in the proof of Proposition 6.1.1, we have\n\n\[ f\left( {S,\bar{S}}\right) = f\left( {S, V}\right) - f\left( {S, S}\right) \]\n\n\[ \underset{\left( \mathrm{F}1\right) }{ = }f\left( {s, V}\right) + \mathop{\sum }\limits_{{v \in S\smallsetminus \{ s\} }}f\left( {v, V}\right) - 0 \]\n\n\[ \underset{\left( {\mathrm{{F2}}}^{\prime }\right) }{ = }f\left( {s, V}\right) \text{.} \] | Yes |
Corollary 6.2.3. In every network (with integral capacity function) there exists an integral flow of maximum total value. | Null | No |
For every multigraph \( G \) there exists a polynomial \( P \) such that, for any finite abelian group \( H \), the number of \( H \) -flows on \( G \) is \( P\left( {\left| H\right| - 1}\right) \) . | Proof. Let \( G = : \left( {V, E}\right) \) ; we use induction on \( m \mathrel{\text{:=}} \left| E\right| \) . Let us assume\n\n(6.1.1)\n\nfirst that all the edges of \( G \) are loops. Then, given any finite abelian group \( H \), every map \( \overrightarrow{E} \rightarrow H \smallsetminus \{ 0\} \) is an \( H \) -flow on \( G \) . Since \( \left| \overrightarrow{E}\right| = \left| E\right| \) when all edges are loops, there are \( {\left( \left| H\right| - 1\right) }^{m} \) such maps, and \( P \mathrel{\text{:=}} {x}^{m} \) is the polynomial sought.\n\nNow assume there is an edge \( {e}_{0} = {xy} \in E \) that is not a loop; let \( {e}_{0} = {xy} \) \( \overrightarrow{{e}_{0}} \mathrel{\text{:=}} \left( {{e}_{0}, x, y}\right) \) and \( {E}^{\prime } \mathrel{\text{:=}} E \smallsetminus \left\{ {e}_{0}\right\} \) . We consider the multigraphs\n\n\[ \n{G}_{1} \mathrel{\text{:=}} G - {e}_{0}\;\text{ and }\;{G}_{2} \mathrel{\text{:=}} G/{e}_{0}.\n\]\n\nBy the induction hypothesis, there are polynomials \( {P}_{i} \) for \( i = 1,2 \) such \( {P}_{1},{P}_{2} \) that, for any finite abelian group \( H \) and \( k \mathrel{\text{:=}} \left| H\right| - 1 \), the number of \( H \) -flows on \( {G}_{i} \) is \( {P}_{i}\left( k\right) \) . We shall prove that the number of \( H \) -flows on \( G \) equals \( {P}_{2}\left( k\right) - {P}_{1}\left( k\right) \) ; then \( P \mathrel{\text{:=}} {P}_{2} - {P}_{1} \) is the desired polynomial.\n\nLet \( H \) be given, and denote the set of all \( H \) -flows on \( G \) by \( F \) . We are trying to show that\n\n\[ \n\left| F\right| = {P}_{2}\left( k\right) - {P}_{1}\left( k\right) \n\]\n\n(1)\n\nThe \( H \) -flows on \( {G}_{1} \) are precisely the restrictions to \( \overrightarrow{{E}^{\prime }} \) of those \( H \) -circulations on \( G \) that are zero on \( {e}_{0} \) but nowhere else. Let us denote the set of these circulations on \( G \) by \( {F}_{1} \) ; then\n\n\[ \n{P}_{1}\left( k\right) = \left| {F}_{1}\right| \n\]\n\nOur aim is to show that, likewise, the \( H \) -flows on \( {G}_{2} \) correspond bijectively to those \( H \) -circulations on \( G \) that are nowhere zero except possibly on \( {e}_{0} \) . The set \( {F}_{2} \) of those circulations on \( G \) then satisfies\n\n\[ \n{P}_{2}\left( k\right) = \left| {F}_{2}\right| \n\]\n\nand \( {F}_{2} \) is the disjoint union of \( {F}_{1} \) and \( F \) . This will prove (1), and hence the theorem. | Yes |
Corollary 6.3.2. If \( H \) and \( {H}^{\prime } \) are two finite abelian groups of equal order, then \( G \) has an \( H \) -flow if and only if \( G \) has an \( {H}^{\prime } \) -flow. | Null | No |
Proposition 6.4.1. A graph has a 2-flow if and only if all its degrees are even. | Proof. By Theorem 6.3.3, a graph \( G = \left( {V, E}\right) \) has a 2-flow if and only if it has a \( {\mathbb{Z}}_{2} \) -flow, i.e. if and only if the constant map \( \overrightarrow{E} \rightarrow {\mathbb{Z}}_{2} \) with value \( \overline{1} \) satisfies (F2). This is the case if and only if all degrees are even. | Yes |
Proposition 6.4.2. A cubic graph has a 3-flow if and only if it is bipartite. | Proof. Let \( G = \left( {V, E}\right) \) be a cubic graph. Let us assume first that \( G \) has a 3-flow, and hence also a \( {\mathbb{Z}}_{3} \) -flow \( f \) . We show that any cycle \( C = {x}_{0}\ldots {x}_{\ell }{x}_{0} \) in \( G \) has even length (cf. Proposition 1.6.1). Consider two consecutive edges on \( C \), say \( {e}_{i - 1} \mathrel{\text{:=}} {x}_{i - 1}{x}_{i} \) and \( {e}_{i} \mathrel{\text{:=}} {x}_{i}{x}_{i + 1} \) . If \( f \) assigned the same value to these edges in the direction of the forward orientation of \( C \), i.e. if \( f\left( {{e}_{i - 1},{x}_{i - 1},{x}_{i}}\right) = f\left( {{e}_{i},{x}_{i},{x}_{i + 1}}\right) \), then \( f \) could not satisfy (F2) at \( {x}_{i} \) for any non-zero value of the third edge at \( {x}_{i} \) . Therefore \( f \) assigns the values \( \overline{1} \) and \( \overline{2} \) to the edges of \( C \) alternately, and in particular \( C \) has even length.\n\nConversely, let \( G \) be bipartite, with vertex bipartition \( \{ X, Y\} \) . Since \( G \) is cubic, the map \( \overrightarrow{E} \rightarrow {\mathbb{Z}}_{3} \) defined by \( f\left( {e, x, y}\right) \mathrel{\text{:=}} \overline{1} \) and \( f\left( {e, y, x}\right) \mathrel{\text{:=}} \overline{2} \) for all edges \( e = {xy} \) with \( x \in X \) and \( y \in Y \) is a \( {\mathbb{Z}}_{3} \) - flow on \( G \) . By Theorem 6.3.3, then, \( G \) has a 3-flow. | Yes |
Proposition 6.4.3. For all even \( n > 4,\varphi \left( {K}^{n}\right) = 3 \) . | Proof. Proposition 6.4.1 implies that \( \varphi \left( {K}^{n}\right) \geq 3 \) for even \( n \) . We show, by induction on \( n \), that every \( G = {K}^{n} \) with even \( n > 4 \) has a 3-flow.\n\nFor the induction start, let \( n = 6 \) . Then \( G \) is the edge-disjoint union of three graphs \( {G}_{1},{G}_{2},{G}_{3} \), with \( {G}_{1},{G}_{2} = {K}^{3} \) and \( {G}_{3} = {K}_{3,3} \) . Clearly \( {G}_{1} \) and \( {G}_{2} \) each have a 2-flow, while \( {G}_{3} \) has a 3-flow by Proposition 6.4.2. The union of all these flows is a 3-flow on \( G \) .\n\nNow let \( n > 6 \), and assume the assertion holds for \( n - 2 \) . Clearly, \( G \) is the edge-disjoint union of a \( {K}^{n - 2} \) and a graph \( {G}^{\prime } = \left( {{V}^{\prime },{E}^{\prime }}\right) \) with \( {G}^{\prime } = \) \( \overline{{K}^{n - 2}} * {K}^{2} \) . The \( {K}^{n - 2} \) has a 3-flow by induction. By Theorem 6.3.3, it thus suffices to find a \( {\mathbb{Z}}_{3} \) -flow on \( {G}^{\prime } \) . For every vertex \( z \) of the \( \overline{{K}^{n - 2}} \subseteq {G}^{\prime } \) , let \( {f}_{z} \) be a \( {\mathbb{Z}}_{3} \) -flow on the triangle \( {zxyz} \subseteq {G}^{\prime } \), where \( e = {xy} \) is the edge of the \( {K}^{2} \) in \( {G}^{\prime } \) . Let \( f : {\overrightarrow{E}}^{\prime } \rightarrow {\mathbb{Z}}_{3} \) be the sum of these flows. Clearly, \( f \) is nowhere zero, except possibly in \( \left( {e, x, y}\right) \) and \( \left( {e, y, x}\right) \) . If \( f\left( {e, x, y}\right) \neq \overline{0} \) , then \( f \) is the desired \( {\mathbb{Z}}_{3} \) -flow on \( {G}^{\prime } \) . If \( f\left( {e, x, y}\right) = \overline{0} \), then \( f + {f}_{z} \) (for any \( z \) ) is a \( {\mathbb{Z}}_{3} \) -flow on \( {G}^{\prime } \) . | Yes |
Proposition 6.4.4. Every 4-edge-connected graph has a 4-flow. | Proof. Let \( G \) be a 4-edge-connected graph. By Corollary 2.4.2, \( G \) has\n\ntwo edge-disjoint spanning trees \( {T}_{i}, i = 1,2 \) . For each edge \( e \notin {T}_{i} \), let \( {C}_{i, e} \) be the unique cycle in \( {T}_{i} + e \), and let \( {f}_{i, e} \) be a \( {\mathbb{Z}}_{4} \) -flow of value \( \bar{i} \) \( {f}_{1, e},{f}_{2, e} \) around \( {C}_{i, e} \) -more precisely: a \( {\mathbb{Z}}_{4} \) -circulation on \( G \) with values \( \bar{i} \) and \( - \bar{i} \) on the edges of \( {C}_{i, e} \) and zero otherwise.\n\nLet \( {f}_{1} \mathrel{\text{:=}} \mathop{\sum }\limits_{{e \notin {T}_{1}}}{f}_{1, e} \) . Since each \( e \notin {T}_{1} \) lies on only one cycle \( {C}_{1,{e}^{\prime }} \) (namely, for \( e = {e}^{\prime } \) ), \( {f}_{1} \) takes only the values \( \overline{1} \) and \( - \overline{1}\left( { = \overline{3}}\right) \) outside \( {T}_{1} \) . Let\n\n\[ F \mathrel{\text{:=}} \left\{ {e \in E\left( {T}_{1}\right) \mid {f}_{1}\left( e\right) = \overline{0}}\right\} \]\n\nand \( {f}_{2} \mathrel{\text{:=}} \mathop{\sum }\limits_{{e \in F}}{f}_{2, e} \) . As above, \( {f}_{2}\left( e\right) = \overline{2} = - \overline{2} \) for all \( e \in F \) . Now \( f \mathrel{\text{:=}} {f}_{1} + {f}_{2} \) is the sum of \( {\mathbb{Z}}_{4} \) -circulations, and hence itself a \( {\mathbb{Z}}_{4} \) -circulation. Moreover, \( f \) is nowhere zero: on edges in \( F \) it takes the value \( \overline{2} \), on edges of \( {T}_{1} - F \) it agrees with \( {f}_{1} \) (and is hence non-zero by the choice of \( F \) ), and on all edges outside \( {T}_{1} \) it takes one of the values \( \overline{1} \) or \( \overline{3} \) . Hence, \( f \) is a \( {\mathbb{Z}}_{4} \) -flow on \( G \), and the assertion follows by Theorem 6.3.3. | Yes |
Corollary 6.4.6. Every cubic 3-edge-colourable graph is bridgeless. | Null | No |
Lemma 6.5.1. There exists a bijection \( {}^{ * } : \overrightarrow{e} \mapsto {\overrightarrow{e}}^{ * } \) from \( \overrightarrow{E} \) to \( \overrightarrow{{E}^{ * }} \) with the following properties:\n\n(i) The underlying edge of \( {\overrightarrow{e}}^{ * } \) is always \( {e}^{ * } \), i.e. \( {\overrightarrow{e}}^{ * } \) is one of the two directions \( \overrightarrow{{e}^{ * }},\overleftarrow{{e}^{ * }} \) of \( {e}^{ * } \) ;\n\n(ii) If \( C \subseteq G \) is a cycle, \( F \mathrel{\text{:=}} E\left( C\right) \), and if \( X \subseteq {V}^{ * } \) is such that \( {F}^{ * } = {E}^{ * }\left( {X,\bar{X}}\right) \), then there exists an orientation \( \overrightarrow{C} \) of \( C \) with \( \left\{ {{\overrightarrow{e}}^{ * } \mid \overrightarrow{e} \in \overrightarrow{C}}\right\} = {\overrightarrow{E}}^{ * }\left( {X,\bar{X}}\right) . | The proof of Lemma 6.5.1 is not entirely trivial: it is based on the so-called orientability of the plane, and we cannot give it here. Still, the assertion of the lemma is intuitively plausible. Indeed if we define for \( e = {vw} \) and \( {e}^{ * } = {xy} \) the assignment \( \left( {e, v, w}\right) \mapsto {\left( e, v, w\right) }^{ * } \in \) \( \left\{ {\left( {{e}^{ * }, x, y}\right) ,\left( {{e}^{ * }, y, x}\right) }\right\} \) simply by turning \( e \) and its ends clockwise onto \( {e}^{ * } \) (Fig. 6.5.1), then the resulting map \( \overrightarrow{e} \mapsto {\overrightarrow{e}}^{ * } \) satisfies the two assertions of the lemma. | No |
For all integers \( r, n \) with \( r > 1 \), every graph \( G \nsupseteq {K}^{r} \) with \( n \) vertices and \( \operatorname{ex}\left( {n,{K}^{r}}\right) \) edges is a \( {T}^{r - 1}\left( n\right) \) . | First proof. We apply induction on \( n \) . For \( n \leq r - 1 \) we have \( G = \) \( {K}^{n} = {T}^{r - 1}\left( n\right) \) as claimed. For the induction step, let now \( n \geq r \) . Since \( G \) is edge-maximal without a \( {K}^{r} \) subgraph, \( G \) has a subgraph \( K = {K}^{r - 1} \) . By the induction hypothesis, \( G - K \) has at most \( {t}_{r - 1}\left( {n - r + 1}\right) \) edges, and each vertex of \( G - K \) has at most \( r - 2 \) neighbours in \( K \) . Hence, \[ \parallel G\parallel \leq {t}_{r - 1}\left( {n - r + 1}\right) + \left( {n - r + 1}\right) \left( {r - 2}\right) + \left( \begin{matrix} r - 1 \\ 2 \end{matrix}\right) = {t}_{r - 1}\left( n\right) ; \] (1) the equality on the right follows by inspection of the Turán graph \( {T}^{r - 1}\left( n\right) \) (Fig. 7.1.3). Since \( G \) is extremal for \( {K}^{r} \) (and \( {T}^{r - 1}\left( n\right) \nsupseteq {K}^{r} \) ), we have equality in (1). Thus, every vertex of \( G - K \) has exactly \( r - 2 \) neighbours in \( K - \) \( {x}_{1},\ldots ,{x}_{r - 1} \) just like the vertices \( {x}_{1},\ldots ,{x}_{r - 1} \) of \( K \) itself. For \( i = 1,\ldots, r - 1 \) let \( {V}_{1},\ldots ,{V}_{r - 1} \) \[ {V}_{i} \mathrel{\text{:=}} \left\{ {v \in V\left( G\right) \mid v{x}_{i} \notin E\left( G\right) }\right\} \] be the set of all vertices of \( G \) whose \( r - 2 \) neighbours in \( K \) are precisely the vertices other than \( {x}_{i} \) . Since \( {K}^{r} \nsubseteq G \), each of the sets \( {V}_{i} \) is independent, and they partition \( V\left( G\right) \) . Hence, \( G \) is \( \left( {r - 1}\right) \) -partite. As \( {T}^{r - 1}\left( n\right) \) is the unique \( \left( {r - 1}\right) \) -partite graph with \( n \) vertices and the maximum number of edges, our claim that \( G = {T}^{r - 1}\left( n\right) \) follows from the assumed extremality of \( G \) . | Yes |
Theorem 7.1.2. (Erdős & Stone 1946)\n\nFor all integers \( r \geq 2 \) and \( s \geq 1 \), and every \( \epsilon > 0 \), there exists an integer \( {n}_{0} \) such that every graph with \( n \geq {n}_{0} \) vertices and at least\n\n\[ \n{t}_{r - 1}\left( n\right) + \epsilon {n}^{2} \n\]\n\nedges contains \( {K}_{s}^{r} \) as a subgraph. | A proof of the Erdős-Stone theorem will be given in Section 7.5, as an illustration of how the regularity lemma may be applied. But the theorem can also be proved directly; see the notes for references. | No |
Corollary 7.1.3. For every graph \( H \) with at least one edge,\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\operatorname{ex}\left( {n, H}\right) {\left( \begin{array}{l} n \\ 2 \end{array}\right) }^{-1} = \frac{\chi \left( H\right) - 2}{\chi \left( H\right) - 1}. \] | Proof of Corollary 7.1.3. Let \( r \mathrel{\text{:=}} \chi \left( H\right) \) . Since \( H \) cannot be coloured with \( r - 1 \) colours, we have \( H \nsubseteq {T}^{r - 1}\left( n\right) \) for all \( n \in \mathbb{N} \), and hence\n\n\[ {t}_{r - 1}\left( n\right) \leq \operatorname{ex}\left( {n, H}\right) . \]\n\nOn the other hand, \( H \subseteq {K}_{s}^{r} \) for all sufficiently large \( s \), so\n\n\[ \operatorname{ex}\left( {n, H}\right) \leq \operatorname{ex}\left( {n,{K}_{s}^{r}}\right) \]\n\nfor all those \( s \) . Let us fix such an \( s \) . For every \( \epsilon > 0 \), Theorem 7.1.2 implies that eventually (i.e. for large enough \( n \) )\n\n\[ \operatorname{ex}\left( {n,{K}_{s}^{r}}\right) < {t}_{r - 1}\left( n\right) + \epsilon {n}^{2}. \]\n\nHence for \( n \) large,\n\n\[ {t}_{r - 1}\left( n\right) /\left( \begin{array}{l} n \\ 2 \end{array}\right) \leq \operatorname{ex}\left( {n, H}\right) /\left( \begin{array}{l} n \\ 2 \end{array}\right) \]\n\n\[ \leq \operatorname{ex}\left( {n,{K}_{s}^{r}}\right) /\left( \begin{array}{l} n \\ 2 \end{array}\right) \]\n\n\[ < {t}_{r - 1}\left( n\right) /\left( \begin{array}{l} n \\ 2 \end{array}\right) + \epsilon {n}^{2}/\left( \begin{array}{l} n \\ 2 \end{array}\right) \]\n\n\[ = {t}_{r - 1}\left( n\right) /\left( \begin{array}{l} n \\ 2 \end{array}\right) + {2\epsilon }/\left( {1 - \frac{1}{n}}\right) \]\n\n\[ \leq {t}_{r - 1}\left( n\right) /\left( \begin{array}{l} n \\ 2 \end{array}\right) + {4\epsilon }\;\left( {\text{ assume }n \geq 2}\right) . \]\n\nTherefore, since \( {t}_{r - 1}\left( n\right) /\left( \begin{array}{l} n \\ 2 \end{array}\right) \) converges to \( \frac{r - 2}{r - 1} \) (Lemma 7.1.4), so does \( \operatorname{ex}\left( {n, H}\right) /\left( \begin{array}{l} n \\ 2 \end{array}\right) \) . | Yes |
Lemma 7.1.4.\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}{t}_{r - 1}\left( n\right) {\left( \begin{array}{l} n \\ 2 \end{array}\right) }^{-1} = \frac{r - 2}{r - 1}. \] | Null | No |
Theorem 7.2.1. There is a constant \( c \in \mathbb{R} \) such that, for every \( r \in \mathbb{N} \), every graph \( G \) of average degree \( d\left( G\right) \geq c{r}^{2} \) contains \( {K}^{r} \) as a topological minor. | Proof. We prove the theorem with \( c = {10} \). Let \( G \) be a graph of average degree at least \( {10}{r}^{2} \). By Theorem 1.4.3 with \( k \mathrel{\text{:=}} {r}^{2}, G \) has an \( {r}^{2} \)-connected subgraph \( H \) with \( \varepsilon \left( H\right) > \varepsilon \left( G\right) - {r}^{2} \geq 4{r}^{2} \). To find a \( T{K}^{r} \) in \( H \), we start by picking \( r \) vertices as branch vertices, and \( r - 1 \) neighbours of each of these as some initial subdividing vertices. These are \( {r}^{2} \) vertices in total, so as \( \delta \left( H\right) \geq \kappa \left( H\right) \geq {r}^{2} \) they can be chosen distinct. Now all that remains is to link up the subdividing vertices in pairs, by disjoint paths in \( H \) corresponding to the edges of the \( {K}^{r} \) of which we wish to find a subdivision. Such paths exist, because \( H \) is \( \frac{1}{2}{r}^{2} \)-linked by Theorem 3.5.3. | Yes |
Lemma 7.2.3. Let \( d, k \in \mathbb{N} \) with \( d \geq 3 \), and let \( G \) be a graph of minimum degree \( \delta \left( G\right) \geq d \) and girth \( g\left( G\right) \geq {8k} + 3 \) . Then \( G \) has a minor \( H \) of minimum degree \( \delta \left( H\right) \geq d{\left( d - 1\right) }^{k} \) . | Proof. Let \( X \subseteq V\left( G\right) \) be maximal with \( d\left( {x, y}\right) > {2k} \) for all \( x, y \in X \) . For each \( x \in X \) put \( {T}_{x}^{0} \mathrel{\text{:=}} \{ x\} \) . Given \( i < {2k} \), assume that we have defined disjoint trees \( {T}_{x}^{i} \subseteq G \) (one for each \( x \in X \) ) whose vertices together are precisely the vertices at distance at most \( i \) from \( X \) in \( G \) . Joining each vertex at distance \( i + 1 \) from \( X \) to a neighbour at distance \( i \), we obtain a similar set of disjoint trees \( {T}_{x}^{i + 1} \) . As every vertex of \( G \) has distance at most \( {2k} \) from \( X \) (by the maximality of \( X \) ), the trees \( {T}_{x} \mathrel{\text{:=}} {T}_{x}^{2k} \) obtained in this way partition the entire vertex set of \( G \) . Let \( H \) be the minor of \( G \) obtained by contracting every \( {T}_{x} \) .\n\nTo prove that \( \delta \left( H\right) \geq d{\left( d - 1\right) }^{k} \), note first that the \( {T}_{x} \) are induced subgraphs of \( G \), because \( \operatorname{diam}{T}_{x} \leq {4k} \) and \( g\left( G\right) > {4k} + 1 \) . Similarly, there is at most one edge in \( G \) between any two trees \( {T}_{x} \) and \( {T}_{y} \) : two such edges, together with the paths joining their ends in \( {T}_{x} \) and \( {T}_{y} \) , would form a cycle of length at most \( {8k} + 2 < g\left( G\right) \) . So all the edges leaving \( {T}_{x} \) are preserved in the contraction.\n\nHow many such edges are there? Note that, for every vertex \( u \in \) \( {T}_{x}^{k - 1} \), all its \( {d}_{G}\left( u\right) \geq d \) neighbours \( v \) also lie in \( {T}_{x} \) : since \( d\left( {v, x}\right) \leq k \) and \( d\left( {x, y}\right) > {2k} \) for every other \( y \in X \), we have \( d\left( {v, y}\right) > k \geq d\left( {v, x}\right) \) , so \( v \) was added to \( {T}_{x} \) rather than to \( {T}_{y} \) when those trees were defined. Therefore \( {T}_{x}^{k} \), and hence also \( {T}_{x} \), has at least \( d{\left( d - 1\right) }^{k - 1} \) leaves. But every leaf of \( {T}_{x} \) sends at least \( d - 1 \) edges away from \( {T}_{x} \), so \( {T}_{x} \) sends at least \( d{\left( d - 1\right) }^{k} \) edges to (distinct) other trees \( {T}_{y} \) . | Yes |
There exists a function \( f : \mathbb{N} \rightarrow \mathbb{N} \) such that every graph of minimum degree at least 3 and girth at least \( f\left( r\right) \) has a \( {K}^{r} \) minor, for all \( r \in \mathbb{N} \) . | We prove the theorem with \( f\left( r\right) \mathrel{\text{:=}} 8\log r + 4\log \log r + c \), for some constant \( c \in \mathbb{R} \) . Let \( k = k\left( r\right) \in \mathbb{N} \) be minimal with \( 3 \cdot {2}^{k} \geq {c}^{\prime }r\sqrt{\log r} \) , where \( {c}^{\prime } \in \mathbb{R} \) is the constant from Theorem 7.2.2. Then for a suitable constant \( c \in \mathbb{R} \) we have \( {8k} + 3 \leq 8\log r + 4\log \log r + c \), and the result follows by Lemma 7.2.3 and Theorem 7.2.2. | Yes |
Proposition 7.3.1. A graph with at least three vertices is edge-maximal without a \( {K}^{4} \) minor if and only if it can be constructed recursively from triangles by pasting \( {}^{4} \) along \( {K}^{2} \) s. | Proof. Recall first that every \( M{K}^{4} \) contains a \( T{K}^{4} \), because \( \Delta \left( {K}^{4}\right) = 3 \) (Proposition 1.7.2); the graphs without a \( {K}^{4} \) minor thus coincide with those without a topological \( {K}^{4} \) minor. The proof that any graph constructible as described is edge-maximal without a \( {K}^{4} \) minor is left as an easy exercise; in order to deduce Hadwiger’s conjecture for \( r = 4 \), we only need the converse implication anyhow. We prove this by induction on \( \left| G\right| \) . Let \( G \) be given, edge-maximal without a \( {K}^{4} \) minor. If \( \left| G\right| = 3 \) then \( G \) is itself a triangle, so let \( \left| G\right| \geq 4 \) for the induction step. Then \( G \) is not complete; let \( S \subseteq V\left( G\right) \) be a separator of size \( \kappa \left( G\right) \), and let \( {C}_{1},{C}_{2} \) be distinct components of \( G - S \) . Since \( S \) is a minimal separator, every vertex in \( S \) has a neighbour in \( {C}_{1} \) and another in \( {C}_{2} \) . If \( \left| S\right| \geq 3 \), this implies that \( G \) contains three independent paths \( {P}_{1},{P}_{2},{P}_{3} \) between a vertex \( {v}_{1} \in {C}_{1} \) and a vertex \( {v}_{2} \in {C}_{2} \) . Since \( \kappa \left( G\right) = \left| S\right| \geq 3 \), the graph \( G - \left\{ {{v}_{1},{v}_{2}}\right\} \) is connected and contains a (shortest) path \( P \) between two different \( {P}_{i} \) . Then \( P \cup {P}_{1} \cup {P}_{2} \cup {P}_{3} = T{K}^{4} \), a contradiction. Hence \( \kappa \left( G\right) \leq 2 \), and the assertion follows from Lemma 4.4. \( {4}^{5} \) and the induction hypothesis. | No |
Corollary 7.3.2. Every edge-maximal graph \( G \) without a \( {K}^{4} \) minor has \( 2\left| G\right| - 3 \) edges. | Proof. Induction on \( \left| G\right| \) . | No |
Corollary 7.3.3. Hadwiger’s conjecture holds for \( r = 4 \) . | Proof. If \( G \) arises from \( {G}_{1} \) and \( {G}_{2} \) by pasting along a complete graph, then \( \chi \left( G\right) = \max \left\{ {\chi \left( {G}_{1}\right) ,\chi \left( {G}_{2}\right) }\right\} \) (see the proof of Proposition 5.5.2). Hence, Proposition 7.3.1 implies by induction on \( \left| G\right| \) that all edge-maximal (and hence all) graphs without a \( {K}^{4} \) minor can be 3-coloured. | No |
Theorem 7.3.4. (Wagner 1937)\n\nLet \( G \) be an edge-maximal graph without a \( {K}^{5} \) minor. If \( \left| G\right| \geq 4 \) then \( G \) can be constructed recursively, by pasting along triangles and \( {K}^{2}s \) , from plane triangulations and copies of the graph \( W \) (Fig. 7.3.1). | Null | No |
Corollary 7.3.5. A graph with \( n \) vertices and no \( {K}^{5} \) minor has at most \( {3n} - 6 \) edges. | Null | No |
Corollary 7.3.9. There is a constant \( g \) such that all graphs \( G \) of girth at least \( g \) satisfy the implication \( \chi \left( G\right) \geq r \Rightarrow G \supseteq T{K}^{r} \) for all \( r \) . | Proof. If \( \chi \left( G\right) \geq r \) then, by Corollary 5.2.3, \( G \) has a subgraph \( H \) of minimum degree \( \delta \left( H\right) \geq r - 1 \) . As \( g\left( H\right) \geq g\left( G\right) \geq g \), Theorem 7.2.5 implies that \( G \supseteq H \supseteq T{K}^{r} \) . | Yes |
Theorem 7.3.7. (Robertson, Seymour & Thomas 1993) Hadwiger’s conjecture holds for \( r = 6 \) . | Null | No |
Theorem 7.3.8. (Kühn & Osthus 2005)\n\nFor every integer \( s \) there is an integer \( {r}_{s} \) such that Hadwiger’s conjecture holds for all graphs \( G \nsupseteq {K}_{s, s} \) and \( r \geq {r}_{s} \) . | Proof. If \( \chi \left( G\right) \geq r \) then, by Corollary 5.2.3, \( G \) has a subgraph \( H \) of minimum degree \( \delta \left( H\right) \geq r - 1 \) . As \( g\left( H\right) \geq g\left( G\right) \geq g \), Theorem 7.2.5 implies that \( G \supseteq H \supseteq T{K}^{r} \) . | No |
Corollary 7.3.9. There is a constant \( g \) such that all graphs \( G \) of girth at least \( g \) satisfy the implication \( \chi \left( G\right) \geq r \Rightarrow G \supseteq T{K}^{r} \) for all \( r \) . | Proof. If \( \chi \left( G\right) \geq r \) then, by Corollary 5.2.3, \( G \) has a subgraph \( H \) of minimum degree \( \delta \left( H\right) \geq r - 1 \) . As \( g\left( H\right) \geq g\left( G\right) \geq g \), Theorem 7.2.5 implies that \( G \supseteq H \supseteq T{K}^{r} \) . | Yes |
Lemma 7.4.3. Let \( \epsilon > 0 \), and let \( C, D \subseteq V \) be disjoint. If \( \left( {C, D}\right) \) is not \( \epsilon \) -regular, then there are partitions \( \mathcal{C} = \left\{ {{C}_{1},{C}_{2}}\right\} \) of \( C \) and \( \mathcal{D} = \left\{ {{D}_{1},{D}_{2}}\right\} \) of \( D \) such that\n\n\[ q\left( {\mathcal{C},\mathcal{D}}\right) \geq q\left( {C, D}\right) + {\epsilon }^{4}\frac{\left| C\right| \left| D\right| }{{n}^{2}}. \] | Proof. Suppose \( \left( {C, D}\right) \) is not \( \epsilon \) -regular. Then there are sets \( {C}_{1} \subseteq C \) and \( {D}_{1} \subseteq D \) with \( \left| {C}_{1}\right| > \epsilon \left| C\right| \) and \( \left| {D}_{1}\right| > \epsilon \left| D\right| \) such that\n\n\[ \left| \eta \right| > \epsilon \]\n\nfor \( \eta \mathrel{\text{:=}} d\left( {{C}_{1},{D}_{1}}\right) - d\left( {C, D}\right) \) . Let \( \mathcal{C} \mathrel{\text{:=}} \left\{ {{C}_{1},{C}_{2}}\right\} \) and \( \mathcal{D} \mathrel{\text{:=}} \left\{ {{D}_{1},{D}_{2}}\right\} \) , where \( {C}_{2} \mathrel{\text{:=}} C \smallsetminus {C}_{1} \) and \( {D}_{2} \mathrel{\text{:=}} D \smallsetminus {D}_{1} \) .\n\nLet us show that \( \mathcal{C} \) and \( \mathcal{D} \) satisfy the conclusion of the lemma. We\n\nshall write \( {c}_{i} \mathrel{\text{:=}} \left| {C}_{i}\right| ,{d}_{i} \mathrel{\text{:=}} \left| {D}_{i}\right| ,{e}_{ij} \mathrel{\text{:=}} \begin{Vmatrix}{{C}_{i},{D}_{j}}\end{Vmatrix}, c \mathrel{\text{:=}} \left| C\right|, d \mathrel{\text{:=}} \left| D\right| \) \( {c}_{i},{d}_{i},{e}_{ij} \) and \( e \mathrel{\text{:=}} \parallel C, D\parallel \) . As in the proof of Lemma 7.4.2, \( c, d, e \)\n\n\[ q\left( {\mathcal{C},\mathcal{D}}\right) = \frac{1}{{n}^{2}}\mathop{\sum }\limits_{{i, j}}\frac{{e}_{ij}^{2}}{{c}_{i}{d}_{j}} \]\n\n\[ = \frac{1}{{n}^{2}}\left( {\frac{{e}_{11}^{2}}{{c}_{1}{d}_{1}} + \mathop{\sum }\limits_{{i + j > 2}}\frac{{e}_{ij}^{2}}{{c}_{i}{d}_{j}}}\right) \]\n\n\[ \underset{\left( 1\right) }{ \geq }\frac{1}{{n}^{2}}\left( {\frac{{e}_{11}^{2}}{{c}_{1}{d}_{1}} + \frac{{\left( e - {e}_{11}\right) }^{2}}{{cd} - {c}_{1}{d}_{1}}}\right) . \]\n\nBy definition of \( \eta \), we have \( {e}_{11} = {c}_{1}{d}_{1}e/{cd} + \eta {c}_{1}{d}_{1} \), so\n\n\[ {n}^{2}q\left( {\mathcal{C},\mathcal{D}}\right) \geq \frac{1}{{c}_{1}{d}_{1}}{\left( \frac{{c}_{1}{d}_{1}e}{cd} + \eta {c}_{1}{d}_{1}\right) }^{2} \]\n\n\[ + \frac{1}{{cd} - {c}_{1}{d}_{1}}{\left( \frac{{cd} - {c}_{1}{d}_{1}}{cd}e - \eta {c}_{1}{d}_{1}\right) }^{2} \]\n\n\[ = \frac{{c}_{1}{d}_{1}{e}^{2}}{{c}^{2}{d}^{2}} + \frac{{2e\eta }{c}_{1}{d}_{1}}{cd} + {\eta }^{2}{c}_{1}{d}_{1} \]\n\n\[ + \frac{{cd} - {c}_{1}{d}_{1}}{{c}^{2}{d}^{2}}{e}^{2} - \frac{{2e\eta }{c}_{1}{d}_{1}}{cd} + \frac{{\eta }^{2}{c}_{1}^{2}{d}_{1}^{2}}{{cd} - {c}_{1}{d}_{1}} \]\n\n\[ \geq \frac{{e}^{2}}{cd} + {\eta }^{2}{c}_{1}{d}_{1} \]\n\n\[ \mathop{\sum }\limits_{\left( 2\right) }\frac{{e}^{2}}{cd} + {\epsilon }^{4}{cd} \]\n\nsince \( {c}_{1} \geq {\epsilon c} \) and \( {d}_{1} \geq {\epsilon d} \) by the choice of \( {C}_{1} \) and \( {D}_{1} \) . | Yes |
Lemma 7.5.1. Let \( \left( {A, B}\right) \) be an \( \epsilon \) -regular pair, of density \( d \) say, and let \( Y \subseteq B \) have size \( \left| Y\right| \geq \epsilon \left| B\right| \) . Then all but fewer than \( \epsilon \left| A\right| \) of the vertices in \( A \) have (each) at least \( \left( {d - \epsilon }\right) \left| Y\right| \) neighbours in \( Y \) . | Proof. Let \( X \subseteq A \) be the set of vertices with fewer than \( \left( {d - \epsilon }\right) \left| Y\right| \) neighbours in \( Y \) . Then \( \parallel X, Y\parallel < \left| X\right| \left( {d - \epsilon }\right) \left| Y\right| \), so\n\n\[ d\left( {X, Y}\right) = \frac{\parallel X, Y\parallel }{\left| X\right| \left| Y\right| } < d - \epsilon = d\left( {A, B}\right) - \epsilon .\n\]\nAs \( \left( {A, B}\right) \) is \( \epsilon \) -regular and \( \left| Y\right| \geq \epsilon \left| B\right| \), this implies that \( \left| X\right| < \epsilon \left| A\right| \) . | Yes |
Proposition 8.1.1. Every connected graph contains a spanning tree. | First proof (by Zorn's lemma).\n\nGiven a connected graph \( G \), consider the set of all trees \( T \subseteq G \), ordered by the subgraph relation. Since \( G \) is connected, any maximal such tree contains every vertex of \( G \), i.e. is a spanning tree of \( G \) .\n\nTo prove that a maximal tree exists, we have to show that for any chain \( \mathcal{C} \) of such trees there is an upper bound: a tree \( {T}^{ * } \subseteq G \) containing every tree in \( \mathcal{C} \) as a subgraph. We claim that \( {T}^{ * } \mathrel{\text{:=}} \bigcup \mathcal{C} \) is such a tree.\n\nTo show that \( {T}^{ * } \) is connected, let \( u, v \in {T}^{ * } \) be two vertices. Then in \( \mathcal{C} \) there is a tree \( {T}_{u} \) containing \( u \) and a tree \( {T}_{v} \) containing \( v \) . One of these is a subgraph of the other, say \( {T}_{u} \subseteq {T}_{v} \) . Then \( {T}_{v} \) contains a path from \( u \) to \( v \), and this path is also contained in \( {T}^{ * } \) .\n\nTo show that \( {T}^{ * } \) is acyclic, suppose it contains a cycle \( C \) . Each of the edges of \( C \) lies in some tree in \( \mathcal{C} \) . These trees form a finite subchain of \( \mathcal{C} \) , which has a maximal element \( T \) . Then \( C \subseteq T \), a contradiction. | Yes |
Let \( G = \left( {V, E}\right) \) be a graph and \( k \in \mathbb{N} \) . If every finite subgraph of \( G \) has chromatic number at most \( k \), then so does \( G \) . | First proof (for \( G \) countable, by the infinity lemma).\n\nLet \( {v}_{0},{v}_{1},\ldots \) be an enumeration of \( V \) and put \( {G}_{n} \mathrel{\text{:=}} G\left\lbrack {{v}_{0},\ldots ,{v}_{n}}\right\rbrack \) . Write \( {V}_{n} \) for the set of all \( k \) -colourings of \( {G}_{n} \) with colours in \( \{ 1,\ldots, k\} \) . Define a graph on \( \mathop{\bigcup }\limits_{{n \in \mathbb{N}}}{V}_{n} \) by inserting all edges \( c{c}^{\prime } \) such that \( c \in {V}_{n} \) and \( {c}^{\prime } \in {V}_{n - 1} \) is the restriction of \( c \) to \( \left\{ {{v}_{0},\ldots ,{v}_{n - 1}}\right\} \) . Let \( {c}_{0}{c}_{1}\ldots \) be a ray in this graph with \( {c}_{n} \in {V}_{n} \) for all \( n \) . Then \( c \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{n \in \mathbb{N}}}{c}_{n} \) is a colouring of \( G \) with colours in \( \{ 1,\ldots, k\} \) . | Yes |
Proposition 8.2.1. Every infinite connected graph has a vertex of infinite degree or contains a ray. | Proof. Let \( G \) be an infinite connected graph with all degrees finite. Let \( {v}_{0} \) be a vertex, and for every \( n \in \mathbb{N} \) let \( {V}_{n} \) be the set of vertices at distance \( n \) from \( {v}_{0} \) . Induction on \( n \) shows that the sets \( {V}_{n} \) are finite, and hence that \( {V}_{n + 1} \neq \varnothing \) (because \( G \) is infinite and connected). Furthermore, the neighbour of a vertex \( v \in {V}_{n + 1} \) on any shortest \( v - {v}_{0} \) path lies in \( {V}_{n} \) . By Lemma 8.1.2, \( G \) contains a ray. | Yes |
Lemma 8.2.2. (Star-Comb Lemma)\n\nLet \( U \) be an infinite set of vertices in a connected graph \( G \) . Then \( G \) contains either a comb with all teeth in \( U \) or a subdivision of an infinite star with all leaves in \( U \) . | Proof. As \( G \) is connected, it contains a path between two vertices in \( U \) . This path is a tree \( T \subseteq G \) every edge of which lies on a path in \( T \) between two vertices in \( U \) . By Zorn’s lemma there is a maximal such tree \( {T}^{ * } \) . Since \( U \) is infinite and \( G \) is connected, \( {T}^{ * } \) is infinite. If \( {T}^{ * } \) has a vertex of infinite degree, it contains the desired subdivided star.\n\nSuppose now that \( {T}^{ * } \) is locally finite. Then \( {T}^{ * } \) contains a ray \( R \) (Proposition 8.2.1). Let us construct a sequence \( {P}_{1},{P}_{2},\ldots \) of disjoint \( R - U \) paths in \( {T}^{ * } \) . Having chosen \( {P}_{i} \) for every \( i < n \) for some \( n \), pick \( v \in R \) so that \( {vR} \) meets none of those paths \( {P}_{i} \) . The first edge of \( {vR} \) lies on a path \( P \) in \( {T}^{ * } \) between two vertices in \( U \) ; let us think of \( P \) as traversing this edge in the same direction as \( R \) . Let \( w \) be the last vertex of \( {vP} \) on \( {vR} \) . Then \( {P}_{n} \mathrel{\text{:=}} {wP} \) contains an \( R - U \) path, and \( {P}_{n} \cap {P}_{i} = \varnothing \) for all \( i < n \) because \( {P}_{i} \cup {Rw} \cup {P}_{n} \) contains no cycle. | Yes |
Lemma 8.2.3. If \( T \) is a normal spanning tree of \( G \), then every end of \( G \) contains exactly one normal ray of \( T \). | Proof. Let \( \omega \in \Omega \left( G\right) \) be given. Apply the star-comb lemma in \( T \) with \( U \) the vertex set of a ray \( R \in \omega \). If the lemma gives a subdivided star with leaves in \( U \) and centre \( z \), say, then the finite down-closure \( \lceil z\rceil \) of \( z \) in \( T \) separates infinitely many vertices \( u > z \) of \( U \) pairwise in \( G \) (Lemma 1.5.5). This contradicts our choice of \( U \). So \( T \) contains a comb with teeth on \( R \). Let \( {R}^{\prime } \subseteq T \) be its spine. Since every ray in \( T \) has an increasing tail (Exercise 4), we may assume that \( {R}^{\prime } \) is a normal ray. Since \( {R}^{\prime } \) is equivalent to \( R \), it lies in \( \omega \). Conversely, distinct normal rays of \( T \) are separated in \( G \) by the (finite) down-closure of their greatest common vertex (Lemma 1.5.5), so they cannot belong to the same end of \( G \). | No |
Every countable connected graph has a normal spanning tree. | Proof. The proof follows that of Proposition 1.5.6; we only sketch the differences. Starting with a single vertex, we construct an infinite sequence \( {T}_{0} \subseteq {T}_{1} \subseteq \ldots \) of finite normal trees in \( G \), all with the same root, whose union \( T \) will be a normal spanning tree. To ensure that \( T \) spans \( G \), we fix an enumeration \( {v}_{0},{v}_{1},\ldots \) of \( V\left( G\right) \) and see to it that \( {T}_{n} \) contains \( {v}_{n} \) . It is clear that \( T \) will be a tree (since any cycle in \( T \) would lie in some \( {T}_{n} \), and every two vertices of \( T \) lie in a common \( {T}_{n} \) and can be linked there), and clearly the tree order of \( T \) induces that of the \( {T}_{n} \) . Finally, \( T \) will be normal, because the endvertices of any edge of \( G \) that is not an edge of \( T \) lie in some \( {T}_{n} \) : since that \( {T}_{n} \) is normal, they must be comparable there, and hence in \( T \) . It remains to specify how to construct \( {T}_{n + 1} \) from \( {T}_{n} \) . If \( {v}_{n + 1} \in {T}_{n} \) , put \( {T}_{n + 1} \mathrel{\text{:=}} {T}_{n} \) . If not, let \( C \) be the component of \( G - {T}_{n} \) containing \( {v}_{n + 1} \) . Let \( x \) be the greatest element of the chain \( N\left( C\right) \) in \( {T}_{n} \), and let \( {T}_{n + 1} \) be the union of \( {T}_{n} \) and an \( x - {v}_{n + 1} \) path \( P \) with \( P \subseteq C \) . Then the neighbourhood in \( {T}_{n + 1} \) of any new component \( {C}^{\prime } \subseteq C \) of \( G - {T}_{n + 1} \) is a chain in \( {T}_{n + 1} \), so \( {T}_{n + 1} \) is again normal. | Yes |
There exists a unique countable graph \( R \) with property \( \left( *\right) \) . | Proof. To prove existence, we construct a graph \( R \) with property \( \left( *\right) \) inductively. Let \( {R}_{0} \mathrel{\text{:=}} {K}^{1} \) . For all \( n \in \mathbb{N} \), let \( {R}_{n + 1} \) be obtained from \( {R}_{n} \) by adding for every set \( U \subseteq V\left( {R}_{n}\right) \) a new vertex \( v \) joined to all the vertices in \( U \) but to none outside \( U \) . (In particular, the new vertices form an independent set in \( \left. {{R}_{n + 1}\text{.}}\right) \) Clearly \( R \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{n \in \mathbb{N}}}{R}_{n} \) has property \( \left( *\right) \) .\n\nTo prove uniqueness, let \( R = \left( {V, E}\right) \) and \( {R}^{\prime } = \left( {{V}^{\prime },{E}^{\prime }}\right) \) be two graphs with property \( \left( *\right) \), each given with a fixed vertex enumeration. We construct a bijection \( \varphi : V \rightarrow {V}^{\prime } \) in an infinite sequence of steps, defining \( \varphi \left( v\right) \) for one new vertex \( v \in V \) at each step.\n\nAt every odd step we look at the first vertex \( v \) in the enumeration of \( V \) for which \( \varphi \left( v\right) \) has not yet been defined. Let \( U \) be the set of those of its neighbours \( u \) in \( R \) for which \( \varphi \left( u\right) \) has already been defined. This is a finite set. Using \( \left( *\right) \) for \( {R}^{\prime } \), find a vertex \( {v}^{\prime } \in {V}^{\prime } \) that is adjacent in \( {R}^{\prime } \) to all the vertices in \( \varphi \left( U\right) \) but to no other vertex in the image of \( \varphi \) (which, so far, is still a finite set). Put \( \varphi \left( v\right) \mathrel{\text{:=}} {v}^{\prime } \) .\n\nAt even steps in the definition process we do the same thing with the roles of \( R \) and \( {R}^{\prime } \) interchanged: we look at the first vertex \( {v}^{\prime } \) in the enumeration of \( {V}^{\prime } \) that does not yet lie in the image of \( \varphi \), and set \( \varphi \left( v\right) = {v}^{\prime } \) for a vertex \( v \) that matches the adjacencies and non-adjacencies of \( {v}^{\prime } \) among the vertices for which \( \varphi \) (resp. \( {\varphi }^{-1} \) ) has already been defined.\n\nBy our minimum choices of \( v \) and \( {v}^{\prime } \), the bijection gets defined on all of \( V \) and all of \( {V}^{\prime } \), and it is clearly an isomorphism. | Yes |
The Rado graph is the only countable graph \( G \) other than \( {K}^{{\aleph }_{0}} \) and \( \overline{{K}^{{\aleph }_{0}}} \) such that, no matter how \( V\left( G\right) \) is partitioned into two parts, one of the parts induces an isomorphic copy of \( G \) . | We first show that the Rado graph \( R \) has the partition property. Let \( \left\{ {{V}_{1},{V}_{2}}\right\} \) be a partition of \( V\left( R\right) \) . If \( \left( *\right) \) fails in both \( R\left\lbrack {V}_{1}\right\rbrack \) and \( R\left\lbrack {V}_{2}\right\rbrack \), say for sets \( {U}_{1},{W}_{1} \) and \( {U}_{2},{W}_{2} \), respectively, then \( \left( *\right) \) fails for \( U = {U}_{1} \cup {U}_{2} \) and \( W = {W}_{1} \cup {W}_{2} \) in \( R \), a contradiction.\n\nTo show uniqueness, let \( G = \left( {V, E}\right) \) be a countable graph with the partition property. Let \( {V}_{1} \) be its set of isolated vertices, and \( {V}_{2} \) the rest. If \( {V}_{1} \neq \varnothing \) then \( G ≄ G\left\lbrack {V}_{2}\right\rbrack \), since \( G \) has isolated vertices but \( G\left\lbrack {V}_{2}\right\rbrack \) does not. Hence \( G = G\left\lbrack {V}_{1}\right\rbrack \simeq \overline{{K}^{{\aleph }_{0}}} \) . Similarly, if \( G \) has a vertex adjacent to all other vertices, then \( G = {K}^{{\aleph }_{0}} \) .\n\nAssume now that \( G \) has no isolated vertex and no vertex joined to all other vertices. If \( G \) is not the Rado graph then there are sets \( U, W \) for which (*) fails in \( G \) ; choose these with \( \left| {U \cup W}\right| \) minimum. Assume first that \( U \neq \varnothing \), and pick \( u \in U \) . Let \( {V}_{1} \) consist of \( u \) and all vertices outside \( U \cup W \) that are not adjacent to \( u \), and let \( {V}_{2} \) contain the remaining vertices. As \( u \) is isolated in \( G\left\lbrack {V}_{1}\right\rbrack \), we have \( G ≄ G\left\lbrack {V}_{1}\right\rbrack \) and hence \( G \simeq G\left\lbrack {V}_{2}\right\rbrack \) . By the minimality of \( \left| {U \cup W}\right| \), there is a vertex \( v \in G\left\lbrack {V}_{2}\right\rbrack - U - W \) that is adjacent to every vertex in \( U \smallsetminus \{ u\} \) and to none in \( W \) . But \( v \) is also adjacent to \( u \), because it lies in \( {V}_{2} \) . So \( U, W \) and \( v \) satisfy \( \left( *\right) \) for \( G \), contrary to assumption.\n\nFinally, assume that \( U = \varnothing \) . Then \( W \neq \varnothing \) . Pick \( w \in W \), and consider the partition \( \left\{ {{V}_{1},{V}_{2}}\right\} \) of \( V \) where \( {V}_{1} \) consists of \( w \) and all its neighbours outside \( W \) . As before, \( G ≄ G\left\lbrack {V}_{1}\right\rbrack \) and hence \( G \simeq G\left\lbrack {V}_{2}\right\rbrack \) . Therefore \( U \) and \( W \smallsetminus \{ w\} \) satisfy \( \left( *\right) \) in \( G\left\lbrack {V}_{2}\right\rbrack \), with \( v \in {V}_{2} \smallsetminus W \) say, and then \( U, W, v \) satisfy \( \left( *\right) \) in \( G \) . | Yes |
Theorem 8.3.3. (Lachlan & Woodrow 1980)\n\nEvery countably infinite homogeneous graph is one of the following:\n\n- a disjoint union of complete graphs of the same order, or the complement of such a graph;\n\n- the graph \( {R}^{r} \) or its complement, for some \( r \geq 3 \) ;\n\n- the Rado graph \( R \) . | Null | No |
There exists a universal planar graph for the minor relation. | Null | No |
Proposition 8.4.1. Let \( G \) be any graph, \( k \in \mathbb{N} \), and let \( A, B \) be two sets of vertices in \( G \) that can be separated by \( k \) but no fewer than \( k \) vertices. Then \( G \) contains \( k \) disjoint \( A - B \) paths. | Proof. By assumption, every set of disjoint \( A - B \) paths has cardinality at most \( k \) . Choose one, \( \mathcal{P} \) say, of maximum cardinality. Suppose \( \left| \mathcal{P}\right| < k \) . Then no set \( X \) consisting of one vertex from each path in \( \mathcal{P} \) separates \( A \) from \( B \) . For each \( X \), let \( {P}_{X} \) be an \( A - B \) path avoiding \( X \) . Let \( H \) be the union of \( \bigcup \mathcal{P} \) with all these paths \( {P}_{X} \) . This is a finite graph in which no set of \( \left| \mathcal{P}\right| \) vertices separates \( A \) from \( B \) . So \( H \subseteq G \) contains more than \( \left| \mathcal{P}\right| \) paths from \( A \) to \( B \) by Menger’s theorem (3.3.1), which contradicts the choice of \( \mathcal{P} \) . | Yes |
Let \( G \) be any graph, and let \( A, B \subseteq V\left( G\right) \) . Then \( G \) contains a set \( \mathcal{P} \) of disjoint \( A - B \) paths and an \( A - B \) separator on \( \mathcal{P} \) . | The next few pages give a proof of Theorem 8.4.2 for countable \( G \) . Of the three proofs we gave for the finite case of Menger's theorem, only the last has any chance of being adaptable to the infinite case: the others were by induction on \( \left| \mathcal{P}\right| \) or on \( \left| G\right| + \parallel G\parallel \), and both these parameters may now be infinite. The third proof, however, looks more promising: recall that, by Lemmas 3.3.2 and 3.3.3, it provided us with a tool to either find a separator on a given system of \( A - B \) paths, or to construct another system of \( A - B \) paths that covers more vertices in \( A \) and in \( B \) . Lemmas 3.3.2 and 3.3.3 (whose proofs work for infinite graphs too) will indeed form a cornerstone of our proof for Theorem 8.4.2. However, it will not do just to apply these lemmas infinitely often. Indeed, although any finite number of applications of Lemma 3.3.2 leaves us with another system of disjoint \( A - B \) paths, an infinite number of iterations may leave nothing at all: each edge may be toggled on and off infinitely often by successive alternating paths, so that no ’limit system’ of \( A - B \) paths will be defined. We shall therefore take another tack: starting at \( A \) , we grow simultaneously as many disjoint paths towards \( B \) as possible. To make this precise, we need some terminology. Given a set \( X \subseteq \) \( V\left( G\right) \), let us write \( {G}_{X \rightarrow B} \) for the subgraph of \( G \) induced by \( X \) and all \( {G}_{X \rightarrow B} \) the components of \( G - X \) that meet \( B \) . Let \( \mathcal{W} = \left( {{W}_{a} \mid a \in A}\right) \) be a family of disjoint paths such that every \( {W}_{a} \) starts in \( a \) . We call \( \mathcal{W} \) an \( A \rightarrow B \) wave in \( G \) if the set \( Z \) of final vertices of paths in \( \mathcal{W} \) separates \( A \) from \( B \) in \( G \) . (Note that \( \mathcal{W} \) may contain infinite paths, which have no final vertex.) Sometimes, we shall wish to consider \( A \rightarrow B \) waves in subgraphs of \( G \) that contain \( A \) but not all of \( B \) . For this reason we do not formally require that \( B \subseteq V\left( G\right) \) . | No |
Lemma 8.4.3. If \( G \) has no proper \( A \rightarrow B \) wave, then \( G \) contains a set of disjoint \( A - B \) paths linking all of \( A \) to \( B \) . | Our approach to the proof of Lemma 8.4.3 is to enumerate the vertices in \( A = : \left\{ {{a}_{1},{a}_{2},\ldots }\right\} \), and to find the required \( A - B \) paths \( {P}_{n} = \)\n\n--- \n\n\( {a}_{1},{a}_{2},\ldots \)\n\n--- \n\n\( {a}_{n}\ldots {b}_{n} \) in turn for \( n = 1,2,\ldots \) . Since our premise in Lemma 8.4.3 is that \( G \) has no proper \( A \rightarrow B \) wave, we would like to choose \( {P}_{1} \) so that \( G - {P}_{1} \) has no proper \( \left( {A \smallsetminus \left\{ {a}_{1}\right\} }\right) \rightarrow B \) wave: this would restore the same premise to \( G - {P}_{1} \), and we could proceed to find \( {P}_{2} \) in \( G - {P}_{1} \) in the same way.\n\nWe shall not be able to choose \( {P}_{1} \) just like this, but we shall be able to do something almost as good. We shall construct \( {P}_{1} \) so that deleting it (as well as a few more vertices outside \( A \) ) leaves a graph that has a large maximal \( \left( {A \smallsetminus \left\{ {a}_{1}\right\} }\right) \rightarrow B \) wave \( \left( {\mathcal{W},{A}^{\prime }}\right) \) . We then earmark the paths \( {W}_{n} = {a}_{n}\ldots {a}_{n}^{\prime }\left( {n \geq 2}\right) \) of this wave as initial segments for the paths \( {P}_{n} \) . By the maximality of \( \mathcal{W} \), there is no proper \( {A}^{\prime } \rightarrow B \) wave in \( {G}_{{A}^{\prime } \rightarrow B} \) . In other words, we have restored our original premise to \( {G}_{{A}^{\prime } \rightarrow B} \) , and can find there an \( {A}^{\prime } - B \) path \( {P}_{2}^{\prime } = {a}_{2}^{\prime }\ldots {b}_{2} \) . Then \( {P}_{2} \mathrel{\text{:=}} {a}_{2}{W}_{2}{a}_{2}^{\prime }{P}_{2}^{\prime } \) is our second path for Lemma 8.4.3, and we continue inductively inside \( {G}_{{A}^{\prime } \rightarrow B} \) .\n\nGiven a set \( \widehat{A} \) of vertices in \( G \), let us call a vertex \( a \notin \widehat{A} \) linkable linkable for \( \left( {G,\widehat{A}, B}\right) \) if \( G - \widehat{A} \) contains an \( a - B \) path \( P \) and a set \( X \supseteq V\left( P\right) \) of vertices such that \( G - X \) has a large maximal \( \widehat{A} \rightarrow B \) wave. (The first such \( a \) we shall be considering will be \( {a}_{1} \), and \( \widehat{A} \) will be the set \( \left. {\left\{ {{a}_{2},{a}_{3},\ldots }\right\} \text{. }}\right) | Yes |
Lemma 8.4.4. Let \( {a}^{ * } \in A \) and \( \widehat{A} \mathrel{\text{:=}} A \smallsetminus \left\{ {a}^{ * }\right\} \), and assume that \( G \) has no proper \( A \rightarrow B \) wave. Then \( {a}^{ * } \) is linkable for \( \left( {G,\widehat{A}, B}\right) \) . | Null | No |
Lemma 8.4.5. Let \( x \) be a vertex in \( G - A \) . If \( G \) has no proper \( A \rightarrow B \) wave but \( G - x \) does, then every \( A \rightarrow B \) wave in \( G - x \) is large. | Proof. Suppose \( G - x \) has a small \( A \rightarrow B \) wave \( \left( {\mathcal{W}, X}\right) \) . Put \( {B}^{\prime } \mathrel{\text{:=}} \) \( X \cup \{ x\} \), and let \( \mathcal{P} \) denote the set of \( A - X \) paths in \( \mathcal{W} \) (Fig. 8.4.3). If \( G \) contains an \( A - {B}^{\prime } \) separator \( S \) on \( \mathcal{P} \), then replacing in \( \mathcal{W} \) every \( P \in \mathcal{P} \)\n\n\n\nFig. 8.4.3. A hypothetical small \( A \rightarrow B \) wave in \( G - x \)\n\nwith its initial segment ending in \( S \) we obtain a small (and hence proper) \( A \rightarrow B \) wave in \( G \), which by assumption does not exist. By Lemmas 3.3.3 and 3.3.2, therefore, \( G \) contains a set \( {\mathcal{P}}^{\prime } \) of disjoint \( A - {B}^{\prime } \) paths exceeding \( \mathcal{P} \) . The set of last vertices of these paths contains \( X \) properly, and hence must be all of \( {B}^{\prime } = X \cup \{ x\} \) . But \( {B}^{\prime } \) separates \( A \) from \( B \) in \( G \), so we can turn \( {\mathcal{P}}^{\prime } \) into an \( A \rightarrow B \) wave in \( G \) by adding as singleton paths any vertices of \( A \) it does not cover. As \( x \) lies on \( {\mathcal{P}}^{\prime } \) but not in \( A \) , this is a proper wave, which by assumption does not exist. | Yes |
Proposition 8.4.6. Let \( G \) be a bipartite graph, with bipartition \( \{ A, B\} \) say. If \( G \) contains a matching of \( A \) and a matching of \( B \), then \( G \) has a 1-factor. | Proof. Let \( H \) be the multigraph on \( V\left( G\right) \) whose edge set is the disjoint union of the two matchings. (Thus, any edge that lies in both matchings becomes a double edge in \( H \) .) Every vertex in \( H \) has degree 1 or 2 . In fact, it is easy to check that every component of \( H \) is an even cycle or an infinite path. Picking every other edge from each component, we obtain a 1-factor of \( G \) . | Yes |
Corollary 8.4.9. A bipartite graph with bipartition \( \{ A, B\} \) contains a matching of \( A \) unless there is a set \( S \subseteq A \) such that \( S \) is not matchable to \( N\left( S\right) \) but \( N\left( S\right) \) is matchable to \( S \) . | Proof. Consider a matching \( M \) and a cover \( U \) as in Theorem 8.4.8. Then \( U \cap B \supseteq N\left( {A \smallsetminus U}\right) \) is matchable to \( A \smallsetminus U \), by the edges of \( M \) . And if \( A \smallsetminus U \) is matchable to \( N\left( {A \smallsetminus U}\right) \), then adding this matching to the edges of \( M \) incident with \( A \cap U \) yields a matching of \( A \) . | No |
Theorem 8.4.11. (Aharoni 1988) \( A \) graph \( G \) has a 1-factor if and only if, for every set \( S \subseteq V\left( G\right) \), the set \( {\mathcal{C}}_{G - S}^{\prime } \) is matchable to \( S \) in \( {G}_{S}^{\prime } \) . | Applied to a finite graph, Theorem 8.4.11 implies Tutte's 1-factor theorem (2.2.1): if \( {\mathcal{C}}_{G - S}^{\prime } \) is not matchable to \( S \) in \( {G}_{S}^{\prime } \), then by the marriage theorem there is a subset \( {S}^{\prime } \) of \( S \) that sends edges to more than \( \left| {S}^{\prime }\right| \) components in \( {\mathcal{C}}_{G - S}^{\prime } \) that are also components of \( G - {S}^{\prime } \), and these components are odd because they are factor-critical. | No |
Corollary 8.4.12. Every graph \( G = \left( {V, E}\right) \) has a set \( S \) of vertices that is matchable to \( {\mathcal{C}}_{G - S}^{\prime } \) in \( {G}_{S}^{\prime } \) and such that every component of \( G - S \) not in \( {\mathcal{C}}_{G - S}^{\prime } \) has a 1-factor. Given any such set \( S \), the graph \( G \) has a 1-factor if and only if \( {\mathcal{C}}_{G - S}^{\prime } \) is matchable to \( S \) in \( {G}_{S}^{\prime } \) . | Proof. Given a pair \( \left( {S, M}\right) \) where \( S \subseteq V \) and \( M \) is a matching of \( S \) in \( {G}_{S}^{\prime } \), and given another such pair \( \left( {{S}^{\prime },{M}^{\prime }}\right) \), write \( \left( {S, M}\right) \leq \left( {{S}^{\prime },{M}^{\prime }}\right) \) if\n\n\[ S \subseteq {S}^{\prime } \subseteq V \smallsetminus \bigcup \left\{ {V\left( C\right) \mid C \in {\mathcal{C}}_{G - S}^{\prime }}\right\} \]\n\nand \( M \subseteq {M}^{\prime } \) . Since \( {\mathcal{C}}_{G - S}^{\prime } \subseteq {\mathcal{C}}_{G - {S}^{\prime }}^{\prime } \) for any such \( S \) and \( {S}^{\prime } \), Zorn’s lemma\n\nimplies that there is a maximal such pair \( \left( {S, M}\right) \) . \( S, M \)\n\nFor the first statement, we have to show that every component \( C \) of \( G - S \) that is not in \( {\mathcal{C}}_{G - S}^{\prime } \) has a 1-factor. If it does not, then by Theorem 8.4.11 there is a set \( T \subseteq V\left( C\right) \) such that \( {\mathcal{C}}_{C - T}^{\prime } \) is not matchable to \( T \) in \( {C}_{T}^{\prime } \) . By Corollary 8.4.9, this means that \( {\mathcal{C}}_{C - T}^{\prime } \) has a subset \( \mathcal{C} \) that is not matchable in \( {C}_{T}^{\prime } \) to the set \( {T}^{\prime } \subseteq T \) of its neighbours, while \( {T}^{\prime } \) is matchable to \( \mathcal{C} \) ; let \( {M}^{\prime } \) be such a matching. Then \( \left( {S, M}\right) < \) \( \left( {S \cup {T}^{\prime }, M \cup {M}^{\prime }}\right) \), contradicting the maximality of \( \left( {S, M}\right) \) .\n\nOf the second statement, only the backward implication is nontrivial. Our assumptions now are that \( {\mathcal{C}}_{G - S}^{\prime } \) is matchable to \( S \) in \( {G}_{S}^{\prime } \) and vice versa (by the choice of \( S \) ), so Proposition 8.4.6 yields that \( {G}_{S}^{\prime } \) has a 1-factor. This defines a matching of \( S \) in \( G \) that picks one vertex \( {x}_{C} \) from every component \( C \in {\mathcal{C}}_{G - S}^{\prime } \) and leaves the other components of \( G - S \) untouched. Adding to this matching a 1-factor of \( C - {x}_{C} \) for every \( C \in {\mathcal{C}}_{G - S}^{\prime } \) and a 1 -factor of every other component of \( G - S \), we obtain the desired 1-factor of \( G \) . | Yes |
Proposition 8.5.1. If \( G \) is connected and locally finite, then \( \left| G\right| \) is a compact Hausdorff space. | Proof. Let \( \mathcal{O} \) be an open cover of \( \left| G\right| \) ; we show that \( \mathcal{O} \) has a finite\n\n(8.1.2)\n\nsubcover. Pick a vertex \( {v}_{0} \in G \), write \( {D}_{n} \) for the (finite) set of vertices at distance \( n \) from \( {v}_{0} \), and put \( {S}_{n} \mathrel{\text{:=}} {D}_{0} \cup \ldots \cup {D}_{n - 1} \) . For every \( v \in {D}_{n} \) , let \( C\left( v\right) \) denote the component of \( G - {S}_{n} \) containing \( v \), and let \( \widehat{C}\left( v\right) \) be\n\n---\n\n9 Except in Exercise 62, we never consider the ends of subgraphs as such.\n\n10 Topologists call \( \left| G\right| \) the Freudenthal compactification of \( G \) .\n\n---\n\nits closure together with all inner points of \( C\left( v\right) - {S}_{n} \) edges. Then \( G\left\lbrack {S}_{n}\right\rbrack \) and these \( \widehat{C}\left( v\right) \) together partition \( \left| G\right| \) .\n\nWe wish to prove that, for some \( n \), each of the sets \( \widehat{C}\left( v\right) \) with \( v \in {D}_{n} \) is contained in some \( O\left( v\right) \in \mathcal{O} \) . For then we can take a finite subcover of \( \mathcal{O} \) for \( G\left\lbrack {S}_{n}\right\rbrack \) (which is compact, being a finite union of edges and vertices), and add to it these finitely many sets \( O\left( v\right) \) to obtain the desired finite subcover for \( \left| G\right| \) .\n\nSuppose there is no such \( n \) . Then for each \( n \) the set \( {V}_{n} \) of vertices \( v \in {D}_{n} \) such that no set from \( \mathcal{O} \) contains \( \widehat{C}\left( v\right) \) is non-empty. Moreover, for every neighbour \( u \in {D}_{n - 1} \) of \( v \in {V}_{n} \) we have \( C\left( v\right) \subseteq C\left( u\right) \) because \( {S}_{n - 1} \subseteq {S}_{n} \), and hence \( u \in {V}_{n - 1} \) ; let \( f\left( v\right) \) be such a vertex \( u \) . By the infinity lemma (8.1.2) there is a ray \( R = {v}_{0}{v}_{1}\ldots \) with \( {v}_{n} \in {V}_{n} \) for all \( n \) . Let \( \omega \) be its end, and let \( O \in \mathcal{O} \) contain \( \omega \) . Since \( O \) is open, it contains a basic open neighbourhood of \( \omega \) : there exist a finite set \( S \subseteq V \) and \( \epsilon > 0 \) such that \( {\widehat{C}}_{\epsilon }\left( {S,\omega }\right) \subseteq O \) . Now choose \( n \) large enough that \( {S}_{n} \) contains \( S \) and all its neighbours. Then \( \widehat{C}\left( {v}_{n}\right) \) lies inside a component of \( G - S \) . As \( C\left( {v}_{n}\right) \) contains \( {v}_{n}R \in \omega \), this component must be \( C\left( {S,\omega }\right) \) . Thus\n\n\[ \widehat{C}\left( {v}_{n}\right) \subseteq C\left( {S,\omega }\right) \subseteq O \in \mathcal{O}, \]\n\ncontradicting the fact that \( {v}_{n} \in {V}_{n} \) . | No |
Theorem 8.5.2. For a connected graph \( G \), the space \( \left| G\right| \) is metrizable if and only if \( G \) has a normal spanning tree. | The proof of Theorem 8.5.2 is indicated in Exercises 30 and 63. | No |
Lemma 8.5.3. If \( X \) is compact and \( {A}_{1},{A}_{2} \) are distinct components of \( X \), then \( X \) is a union of disjoint open sets \( {X}_{1},{X}_{2} \) such that \( {A}_{1} \subseteq {X}_{1} \) and \( {A}_{2} \subseteq {X}_{2} \) . | Null | No |
Lemma 8.5.4. If \( G \) is a locally finite graph, then every closed connected subspace of \( \left| G\right| \) is arc-connected. | The proof of Lemma 8.5.4 is not easy; see the notes for a reference. | No |
Lemma 8.5.5. Let \( G \) be connected and locally finite, \( \{ X, Y\} \) a partition of \( V\left( G\right) \), and \( F \mathrel{\text{:=}} E\left( {X, Y}\right) \) . (i) \( F \) is finite if and only if \( \bar{X} \cap \bar{Y} = \varnothing \) . (ii) If \( F \) is finite, there is no arc in \( \left| G\right| \smallsetminus \overset{ \circ }{F} \) with one endpoint in \( X \) and the other in \( Y \) . (iii) If \( F \) is infinite and \( X \) and \( Y \) are both connected in \( G \), there is such an arc. | Proof. (i) Suppose first that \( F \) is infinite. Since \( G \) is locally finite, the set \( {X}^{\prime } \) of endvertices of \( F \) in \( X \) is also infinite. By the star-comb lemma (8.2.2), there is a comb in \( G \) with teeth in \( {X}^{\prime } \) ; let \( \omega \) be the end of its spine. Then every basic open neighbourhood \( {\widehat{C}}_{\epsilon }\left( {S,\omega }\right) \) of \( \omega \) meets \( {X}^{\prime } \) infinitely and hence also meets \( Y \), giving \( \omega \in \bar{X} \cap \bar{Y} \) . Suppose now that \( F \) is finite. Let \( S \) be the set of vertices incident with edges in \( F \) . Then \( S \) is finite and separates \( X \) from \( Y \) . Since every basic open set of the form \( {\widehat{C}}_{\epsilon }\left( {S,\omega }\right) \) misses \( X \) or \( Y \), no end \( \omega \) lies in the closure of both. (ii) Clearly \( \left| G\right| \smallsetminus \overset{ \circ }{F} = \overline{G\left\lbrack X\right\rbrack } \cup \overline{G\left\lbrack Y\right\rbrack } \), and by (i) this union is disjoint. Hence no connected subset of \( \left| G\right| \smallsetminus F \) can meet both \( X \) and \( Y \) , but arcs are continuous images of \( \left\lbrack {0,1}\right\rbrack \) and hence connected. (iii) By (i), there is an end \( \omega \in \bar{X} \cap \bar{Y} \) . Apply the star-comb lemma in \( G\left\lbrack X\right\rbrack \) to any sequence of vertices in \( X \) converging to \( \omega \) ; this yields a comb whose spine \( R \) lies in \( \omega \) . Similarly, there is a comb in \( G\left\lbrack Y\right\rbrack \) whose spine \( {R}^{\prime } \) lies in \( \omega \) . Now \( R \cup \{ \omega \} \cup {R}^{\prime } \) is the desired arc. | Yes |
Lemma 8.5.6. Let \( G \) be locally finite. A closed standard subspace \( C \) of \( \left| G\right| \) is a circle in \( \left| G\right| \) if and only if \( C \) is connected, every vertex in \( C \) is incident with exactly two edges in \( C \), and every end in \( C \) has vertex-degree 2 (equivalently: edge-degree 2) in \( C \) . | It is not hard to show that every circle \( C \) in a space \( \left| G\right| \) is a standard subspace; the set \( D \) of edges it contains will be called its circuit. Then circuit \( C \) is the closure of the point set \( \bigcup D \), as every neighbourhood in \( C \) of a vertex or end meets an edge, which must then be contained in \( C \) and hence lie in \( D \) . In particular, there are no circles consisting only of ends, and every circle is uniquely determined by its circuit. | No |
Corollary 8.5.9. \( \mathcal{C}\left( G\right) \) is generated by finite circuits, and is closed under infinite (thin) sums. | Proof. By Theorem 8.2.4, \( G \) has a normal spanning tree, \( T \) say. By\n\n(8.2.4)\n\nLemma 8.5.7, its closure \( \bar{T} \) in \( \left| G\right| \) is a topological spanning tree. The fundamental circuits of \( \bar{T} \) coincide with those of \( T \), and are therefore finite. By Theorem 8.5.8 (iii), they generate \( \mathcal{C}\left( G\right) \) .\n\nLet \( \mathop{\sum }\limits_{{i \in I}}{D}_{i} \) be a sum of elements of \( \mathcal{C}\left( G\right) \) . By Theorem 8.5.8 (ii), each \( {D}_{i} \) is a disjoint union of circuits. Together, these form a thin family, whose sum equals \( \mathop{\sum }\limits_{{i \in I}}{D}_{i} \) and lies in \( \mathcal{C}\left( G\right) \) . | Yes |
Theorem 8.5.10. The following statements are equivalent for all \( k \in \mathbb{N} \) \( k \) and locally finite multigraphs \( G \) : \( G \)\n\n(i) \( G \) has \( k \) edge-disjoint topological spanning trees.\n\n(ii) For every finite partition of \( V\left( G\right) \), into \( \ell \) sets say, \( G \) has at least \( k\left( {\ell - 1}\right) \) cross-edges. | We begin our proof of Theorem 8.5.10 with a compactness extension of the finite theorem, which will give us a slightly weaker statement at the limit. Following Tutte, let us call a spanning submultigraph \( H \) of \( G \)\n\n---semiconnected in \( G \) if every finite cut of \( G \) contains an edge of \( H \) .\n\nLemma 8.5.11. If for every finite partition of \( V\left( G\right) \), into \( \ell \) sets say, \( G \) has at least \( k\left( {\ell - 1}\right) \) cross-edges, then \( G \) has \( k \) edge-disjoint semiconnected spanning subgraphs.\n\nProof. Pick an enumeration \( {v}_{0},{v}_{1},\ldots \) of \( V\left( G\right) \) . For every \( n \in \mathbb{N} \) let \( {G}_{n} \n\n(8.1.2)\n\nbe the finite multigraph obtained from \( G \) by contracting every component of \( G - \left\{ {{v}_{0},\ldots ,{v}_{n}}\right\} \) to a vertex, deleting any loops but no parallel edges that arise in the contraction. Then \( G\left\lbrack {{v}_{0},\ldots ,{v}_{n}}\right\rbrack \) is an induced submultigraph of \( {G}_{n} \) . Let \( {\mathcal{V}}_{n} \) denote the set of all \( k \) -tuples \( \left( {{H}_{n}^{1},\ldots ,{H}_{n}^{k}}\right) \) of edge-disjoint connected spanning subgraphs of \( {G}_{n} \).\n\nSince every partition \( P \) of \( V\left( {G}_{n}\right) \) induces a partition of \( V\left( G\right) \), since \( G \) has enough cross-edges for that partition, and since all these cross-edges are also cross-edges of \( P \), Theorem 2.4.1 implies that \( {\mathcal{V}}_{n} \neq \varnothing \) . Since every \( \left( {{H}_{n}^{1},\ldots ,{H}_{n}^{k}}\right) \in {\mathcal{V}}_{n} \) induces an element \( \left( {{H}_{n - 1}^{1},\ldots ,{H}_{n - 1}^{k}}\right) \) of \( {\mathcal{V}}_{n - 1} \), the infinity lemma (8.1.2), yields a sequence \( {\left( {H}_{n}^{1},\ldots ,{H}_{n}^{k}\right) }_{n \in \mathbb{N}} \) of \( k \) -tuples, one from each \( {\mathcal{V}}_{n} \), with a limit \( \left( {{H}^{1},\ldots ,{H}^{k}}\right) \) defined by the nested unions\n\n\[{H}^{i} \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{n \in \mathbb{N}}}{H}_{n}^{i}\left\lbrack {{v}_{0},\ldots ,{v}_{n}}\right\rbrack\]\n\nThese \( {H}^{i} \) are edge-disjoint for distinct \( i \) (because the \( {H}_{n}^{i} \) are), but they need not be connected. To show that they are semiconnected in \( G \) , consider a finite cut \( F \) of \( G \) . Choose \( n \) large enough that all the end-vertices of edges in \( F \) are among \( {v}_{0},\ldots ,{v}_{n} \) . Then \( F \) is also a cut of \( {G}_{n} \) . Now consider the \( k \) -tuple \( \left( {{H}_{n}^{1},\ldots ,{H}_{n}^{k}}\right) \) which the infinity lemma picked from \( {\mathcal{V}}_{n} \) . Each of these \( {H}_{n}^{i} \) is a connected spanning subgraph of \( {G}_{n} \), so it contains an edge from \( F \) . But \( {H}_{n}^{i} \) agrees with \( {H}^{i} \) on \( \left\{ {{v}_{0},\ldots ,{v}_{n}}\right\} \), so \( {H}^{i} \) too contains this edge from \( F \) . | Yes |
Lemma 8.5.11. If for every finite partition of \( V\left( G\right) \), into \( \ell \) sets say, \( G \) has at least \( k\left( {\ell - 1}\right) \) cross-edges, then \( G \) has \( k \) edge-disjoint semicon-nected spanning subgraphs. | Proof. Pick an enumeration \( {v}_{0},{v}_{1},\ldots \) of \( V\left( G\right) \) . For every \( n \in \mathbb{N} \) let \( {G}_{n} \) be the finite multigraph obtained from \( G \) by contracting every component of \( G - \left\{ {{v}_{0},\ldots ,{v}_{n}}\right\} \) to a vertex, deleting any loops but no parallel edges that arise in the contraction. Then \( G\left\lbrack {{v}_{0},\ldots ,{v}_{n}}\right\rbrack \) is an induced submultigraph of \( {G}_{n} \) . Let \( {\mathcal{V}}_{n} \) denote the set of all \( k \) -tuples \( \left( {{H}_{n}^{1},\ldots ,{H}_{n}^{k}}\right) \) of edge-disjoint connected spanning subgraphs of \( {G}_{n} \) . Since every partition \( P \) of \( V\left( {G}_{n}\right) \) induces a partition of \( V\left( G\right) \), since \( G \) has enough cross-edges for that partition, and since all these cross-edges are also cross-edges of \( P \), Theorem 2.4.1 implies that \( {\mathcal{V}}_{n} \neq \varnothing \) . Since every \( \left( {{H}_{n}^{1},\ldots ,{H}_{n}^{k}}\right) \in {\mathcal{V}}_{n} \) induces an element \( \left( {{H}_{n - 1}^{1},\ldots ,{H}_{n - 1}^{k}}\right) \) of \( {\mathcal{V}}_{n - 1} \), the infinity lemma (8.1.2), yields a sequence \( {\left( {H}_{n}^{1},\ldots ,{H}_{n}^{k}\right) }_{n \in \mathbb{N}} \) of \( k \) -tuples, one from each \( {\mathcal{V}}_{n} \), with a limit \( \left( {{H}^{1},\ldots ,{H}^{k}}\right) \) defined by the nested unions \[ {H}^{i} \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{n \in \mathbb{N}}}{H}_{n}^{i}\left\lbrack {{v}_{0},\ldots ,{v}_{n}}\right\rbrack \] These \( {H}^{i} \) are edge-disjoint for distinct \( i \) (because the \( {H}_{n}^{i} \) are), but they need not be connected. To show that they are semiconnected in \( G \) , consider a finite cut \( F \) of \( G \) . Choose \( n \) large enough that all the end-vertices of edges in \( F \) are among \( {v}_{0},\ldots ,{v}_{n} \) . Then \( F \) is also a cut of \( {G}_{n} \) . Now consider the \( k \) -tuple \( \left( {{H}_{n}^{1},\ldots ,{H}_{n}^{k}}\right) \) which the infinity lemma picked from \( {\mathcal{V}}_{n} \) . Each of these \( {H}_{n}^{i} \) is a connected spanning subgraph of \( {G}_{n} \), so it contains an edge from \( F \) . But \( {H}_{n}^{i} \) agrees with \( {H}^{i} \) on \( \left\{ {{v}_{0},\ldots ,{v}_{n}}\right\} \), so \( {H}^{i} \) too contains this edge from \( F \) . | Yes |
Lemma 8.5.12. A spanning subgraph \( H \subseteq G \) is semiconnected in \( G \) if and only if its closure \( \bar{H} \) in \( \left| G\right| \) is topologically connected. | Proof. If \( \bar{H} \) is disconnected, it is contained in the union of two closed subsets \( {O}_{1},{O}_{2} \) of \( \left| G\right| \) that both meet \( \bar{H} \) and satisfy \( {O}_{1} \cap {O}_{2} \cap \bar{H} = \varnothing \) . Since \( \bar{H} \) is a standard subspace containing \( V\left( G\right) \), the sets \( {O}_{i} \) partition \( V\left( G\right) \) into two non-empty sets \( {X}_{1},{X}_{2} \) . Then\n\n\[ \n{\bar{X}}_{1} \cap {\bar{X}}_{2} \subseteq {O}_{1} \cap {O}_{2} \cap \Omega \left( G\right) \subseteq {O}_{1} \cap {O}_{2} \cap \bar{H} = \varnothing .\n\]\n\nBy Lemma 8.5.5 (i), this implies that \( G \) has only finitely many \( {X}_{1} - {X}_{2} \) edges. As edges are connected, none of them can lie in \( H \) . Hence, \( H \) is not semiconnected.\n\nThe converse implication is straightforward (and not needed in our proof of Theorem 8.5.10): a finite cut of \( G \) containing no edge of \( H \) defines a partition of \( \bar{H} \) into non-empty open subsets, showing that \( \bar{H} \) is disconnected. | Yes |
Lemma 8.5.13. Every closed, connected, standard subspace \( X \) of \( \left| G\right| \) that contains \( V\left( G\right) \) also contains a topological spanning tree of \( G \) . | Proof. By Lemma 8.5.4, \( X \) is arc-connected. Since \( X \) contains all vertices, \( G \) cannot be disconnected, so its local finiteness implies that it is countable. Let \( {e}_{0},{e}_{1},\ldots \) be an enumeration of the edges in \( X \) .\n\nWe now delete these edges one by one, keeping \( X \) arc-connected. Starting with \( {X}_{0} \mathrel{\text{:=}} X \), we define \( {X}_{n + 1} \mathrel{\text{:=}} {X}_{n} \smallsetminus {e}_{n} \) if this keeps \( {X}_{n + 1} \) arc-connected; if not, we put \( {X}_{n + 1} \mathrel{\text{:=}} {X}_{n} \) . Finally, we put \( T \mathrel{\text{:=}} \mathop{\bigcap }\limits_{{n \in \mathbb{N}}}{X}_{n} \) .\n\nClearly, \( T \) is closed, contains every vertex and every end of \( G \), but contains no circle: any circle in \( T \) would contain an edge, which should have got deleted. To show that \( T \) is arc-connected, it suffices by Lemmas 8.5.4 and 8.5.12 to check that every finite cut of \( G \) contains an edge from \( T \) . By Lemma 8.5.5 (ii), the edges in such a cut could not all be deleted, so one of them lies in \( T \) . | Yes |
For every \( r \in \mathbb{N} \) there exists an \( n \in \mathbb{N} \) such that every graph of order at least \( n \) contains either \( {K}^{r} \) or \( \overline{{K}^{r}} \) as an induced subgraph. | The assertion is trivial for \( r \leq 1 \) ; we assume that \( r \geq 2 \) . Let \( n \mathrel{\text{:=}} {2}^{{2r} - 3} \), and let \( G \) be a graph of order at least \( n \) . We shall define a sequence \( {V}_{1},\ldots ,{V}_{{2r} - 2} \) of sets and choose vertices \( {v}_{i} \in {V}_{i} \) with the following properties:\n\n(i) \( \left| {V}_{i}\right| = {2}^{{2r} - 2 - i}\;\left( {i = 1,\ldots ,{2r} - 2}\right) \) ;\n\n(ii) \( {V}_{i} \subseteq {V}_{i - 1} \smallsetminus \left\{ {v}_{i - 1}\right\} \;\left( {i = 2,\ldots ,{2r} - 2}\right) \) ;\n\n(iii) \( {v}_{i - 1} \) is adjacent either to all vertices in \( {V}_{i} \) or to no vertex in \( {V}_{i} \) \( \left( {i = 2,\ldots ,{2r} - 2}\right) \) .\n\nLet \( {V}_{1} \subseteq V\left( G\right) \) be any set of \( {2}^{{2r} - 3} \) vertices, and pick \( {v}_{1} \in {V}_{1} \) arbitrarily. Then (i) holds for \( i = 1 \), while (ii) and (iii) hold trivially. Suppose now that \( {V}_{i - 1} \) and \( {v}_{i - 1} \in {V}_{i - 1} \) have been chosen so as to satisfy (i)-(iii) for \( i - 1 \), where \( 1 < i \leq {2r} - 2 \) . Since\n\n\[ \left| {{V}_{i - 1} \smallsetminus \left\{ {v}_{i - 1}\right\} }\right| = {2}^{{2r} - 1 - i} - 1 \]\n\nis odd, \( {V}_{i - 1} \) has a subset \( {V}_{i} \) satisfying (i)-(iii); we pick \( {v}_{i} \in {V}_{i} \) arbitrarily.\n\nAmong the \( {2r} - 3 \) vertices \( {v}_{1},\ldots ,{v}_{{2r} - 3} \), there are \( r - 1 \) vertices that show the same behaviour when viewed as \( {v}_{i - 1} \) in (iii), being adjacent either to all the vertices in \( {V}_{i} \) or to none. Accordingly, these \( r - 1 \) vertices and \( {v}_{{2r} - 2} \) induce either a \( {K}^{r} \) or a \( \overline{{K}^{r}} \) in \( G \), because \( {v}_{i},\ldots ,{v}_{{2r} - 2} \in {V}_{i} \) for all \( i \) . | Yes |
Theorem 9.1.2. Let \( k, c \) be positive integers, and \( X \) an infinite set. If \( {\left\lbrack X\right\rbrack }^{k} \) is coloured with \( c \) colours, then \( X \) has an infinite monochromatic subset. | Proof. We prove the theorem by induction on \( k \), with \( c \) fixed. For \( k = 1 \) the assertion holds, so let \( k > 1 \) and assume the assertion for smaller values of \( k \) . Let \( {\left\lbrack X\right\rbrack }^{k} \) be coloured with \( c \) colours. We shall construct an infinite sequence \( {X}_{0},{X}_{1},\ldots \) of infinite subsets of \( X \) and choose elements \( {x}_{i} \in {X}_{i} \) with the following properties (for all \( i \) ): (i) \( {X}_{i + 1} \subseteq {X}_{i} \smallsetminus \left\{ {x}_{i}\right\} \) (ii) all \( k \) -sets \( \left\{ {x}_{i}\right\} \cup Z \) with \( Z \in {\left\lbrack {X}_{i + 1}\right\rbrack }^{k - 1} \) have the same colour, which we associate with \( {x}_{i} \) . We start with \( {X}_{0} \mathrel{\text{:=}} X \) and pick \( {x}_{0} \in {X}_{0} \) arbitrarily. By assumption, \( {X}_{0} \) is infinite. Having chosen an infinite set \( {X}_{i} \) and \( {x}_{i} \in {X}_{i} \) for some \( i \) , we \( c \) -colour \( {\left\lbrack {X}_{i} \smallsetminus \left\{ {x}_{i}\right\} \right\rbrack }^{k - 1} \) by giving each set \( Z \) the colour of \( \left\{ {x}_{i}\right\} \cup Z \) from our \( c \) -colouring of \( {\left\lbrack X\right\rbrack }^{k} \) . By the induction hypothesis, \( {X}_{i} \smallsetminus \left\{ {x}_{i}\right\} \) has an infinite monochromatic subset, which we choose as \( {X}_{i + 1} \) . Clearly, this choice satisfies (i) and (ii). Finally, we pick \( {x}_{i + 1} \in {X}_{i + 1} \) arbitrarily. Since \( c \) is finite, one of the \( c \) colours is associated with infinitely many \( {x}_{i} \) . These \( {x}_{i} \) form an infinite monochromatic subset of \( X \) . | Yes |
Theorem 9.1.3. For all \( k, c, r \geq 1 \) there exists an \( n \geq k \) such that every \( n \) -set \( X \) has a monochromatic \( r \) -subset with respect to any \( c \) -colouring of \( {\left\lbrack X\right\rbrack }^{k} \) . | Proof. As is customary in set theory, we denote by \( n \in \mathbb{N} \) (also) the \( k, c, r \) set \( \{ 0,\ldots, n - 1\} \) . Suppose the assertion fails for some \( k, c, r \) . Then for every \( n \geq k \) there exist an \( n \) -set, without loss of generality the set \( n \), and a \( c \) -colouring \( {\left\lbrack n\right\rbrack }^{k} \rightarrow c \) such that \( n \) contains no monochromatic \( r \) -set. Let us call such colourings bad; we are thus assuming that for every \( n \geq k \) there exists a bad colouring of \( {\left\lbrack n\right\rbrack }^{k} \) . Our aim is to combine these into a bad colouring of \( {\left\lbrack \mathbb{N}\right\rbrack }^{k} \), which will contradict Theorem 9.1.2.\n\nFor every \( n \geq k \) let \( {V}_{n} \neq \varnothing \) be the set of bad colourings of \( {\left\lbrack n\right\rbrack }^{k} \) . For \( n > k \), the restriction \( f\left( g\right) \) of any \( g \in {V}_{n} \) to \( {\left\lbrack n - 1\right\rbrack }^{k} \) is still bad, and hence lies in \( {V}_{n - 1} \) . By the infinity lemma (8.1.2), there is an infinite sequence \( {g}_{k},{g}_{k + 1},\ldots \) of bad colourings \( {g}_{n} \in {V}_{n} \) such that \( f\left( {g}_{n}\right) = {g}_{n - 1} \) for all \( n > k \) . For every \( m \geq k \), all colourings \( {g}_{n} \) with \( n \geq m \) agree on \( {\left\lbrack m\right\rbrack }^{k} \), so for each \( Y \in {\left\lbrack \mathbb{N}\right\rbrack }^{k} \) the value of \( {g}_{n}\left( Y\right) \) coincides for all \( n > \max Y \) . Let us define \( g\left( Y\right) \) as this common value \( {g}_{n}\left( Y\right) \) . Then \( g \) is a bad colouring of \( {\left\lbrack \mathbb{N}\right\rbrack }^{k} \) : every \( r \) -set \( S \subseteq \mathbb{N} \) is contained in some sufficiently large \( n \) , so \( S \) cannot be monochromatic since \( g \) coincides on \( {\left\lbrack n\right\rbrack }^{k} \) with the bad colouring \( {g}_{n} \) . | Yes |
Proposition 9.2.1. Let \( s, t \) be positive integers, and let \( T \) be a tree of order \( t \) . Then \( R\left( {T,{K}^{s}}\right) = \left( {s - 1}\right) \left( {t - 1}\right) + 1 \) . | Proof. The disjoint union of \( s - 1 \) graphs \( {K}^{t - 1} \) contains no copy of \( T \), while the complement of this graph, the complete \( \left( {s - 1}\right) \) -partite graph \( {K}_{t - 1}^{s - 1} \), does not contain \( {K}^{s} \) . This proves \( R\left( {T,{K}^{s}}\right) \geq \left( {s - 1}\right) \left( {t - 1}\right) + 1 \). Conversely, let \( G \) be any graph of order \( n = \left( {s - 1}\right) \left( {t - 1}\right) + 1 \) whose complement contains no \( {K}^{s} \) . Then \( s > 1 \), and in any vertex colouring of \( G \) (in the sense of Chapter 5) at most \( s - 1 \) vertices can have the same colour. Hence, \( \chi \left( G\right) \geq \lceil n/\left( {s - 1}\right) \rceil = t \) . By Corollary 5.2.3, \( G \) has a subgraph \( H \) with \( \delta \left( H\right) \geq t - 1 \), which by Corollary 1.5.4 contains a copy of \( T \) . | Yes |
Proposition 9.2.3. If \( T \) is a tree but not a star, then infinitely many graphs are Ramsey-minimal for \( T \) . | Proof. Let \( \left| T\right| = : r \) . We show that for every \( n \in \mathbb{N} \) there is a graph of order at least \( n \) that is Ramsey-minimal for \( T \) .\n\nBy Theorem 5.2.5, there exists a graph \( G \) with chromatic number \( \chi \left( G\right) > {r}^{2} \) and girth \( g\left( G\right) > n \) . If we colour the edges of \( G \) red and green, then the red and the green subgraph cannot both have an \( r \) - (vertex-)colouring in the sense of Chapter 5: otherwise we could colour the vertices of \( G \) with the pairs of colours from those colourings and obtain a contradiction to \( \chi \left( G\right) > {r}^{2} \) . So let \( {G}^{\prime } \subseteq G \) be monochromatic with \( \chi \left( {G}^{\prime }\right) > r \) . By Corollary 5.2.3, \( {G}^{\prime } \) has a subgraph of minimum degree at least \( r \), which contains a copy of \( T \) by Corollary 1.5.4.\n\nLet \( {G}^{ * } \subseteq G \) be Ramsey-minimal for \( T \) . Clearly, \( {G}^{ * } \) is not a forest: the edges of any forest can be 2-coloured (partitioned) so that no monochromatic subforest contains a path of length 3 , let alone a copy of \( T \) . (Here we use that \( T \) is not a star, and hence contains a \( {P}^{3} \) .) So \( {G}^{ * } \) contains a cycle, which has length \( g\left( G\right) > n \) since \( {G}^{ * } \subseteq G \) . In particular, \( \left| {G}^{ * }\right| > n \) as desired. | Yes |
Theorem 9.3.1. Every graph has a Ramsey graph. In other words, for every graph \( H \) there exists a graph \( G \) that, for every partition \( \left\{ {{E}_{1},{E}_{2}}\right\} \) of \( E\left( G\right) \), has an induced subgraph \( H \) with \( E\left( H\right) \subseteq {E}_{1} \) or \( E\left( H\right) \subseteq {E}_{2} \) . | First proof. In our construction of the desired Ramsey graph we shall repeatedly replace vertices of a graph \( G = \left( {V, E}\right) \) already constructed\n\nby copies of another graph \( H \) . For a vertex set \( U \subseteq V \) let \( G\left\lbrack {U \rightarrow H}\right\rbrack \)\n\n\( G\left\lbrack {U \rightarrow H}\right\rbrack \)\n\ndenote the graph obtained from \( G \) by replacing the vertices \( u \in U \) with\n\ncopies \( H\left( u\right) \) of \( H \) and joining each \( H\left( u\right) \) completely to all \( H\left( {u}^{\prime }\right) \) with\n\n\( H\left( u\right) \)\n\n\( u{u}^{\prime } \in E \) and to all vertices \( v \in V \smallsetminus U \) with \( {uv} \in E \) (Fig. 9.3.1). Formally,\n\n\n\nFig. 9.3.1. A graph \( G\left\lbrack {U \rightarrow H}\right\rbrack \) with \( H = {K}^{3} \)\n\n\( G\left\lbrack {U \rightarrow H}\right\rbrack \) is the graph on\n\n\[ \left( {U \times V\left( H\right) }\right) \cup \left( {\left( {V \smallsetminus U}\right) \times \{ \varnothing \} }\right) \]\n\nin which two vertices \( \left( {v, w}\right) \) and \( \left( {{v}^{\prime },{w}^{\prime }}\right) \) are adjacent if and only if either \( v{v}^{\prime } \in E \), or else \( v = {v}^{\prime } \in U \) and \( w{w}^{\prime } \in E\left( H\right) {.}^{3} \) | Yes |
Lemma 9.3.2. Every bipartite graph can be embedded in a bipartite graph of the form \( \left( {X,{\left\lbrack X\right\rbrack }^{k}, E}\right) \) with \( E = \{ {xY} \mid x \in Y\} \) . | Proof. Let \( P \) be any bipartite graph, with vertex classes \( \left\{ {{a}_{1},\ldots ,{a}_{n}}\right\} \) and \( \left\{ {{b}_{1},\ldots ,{b}_{m}}\right\} \), say. Let \( X \) be a set with \( {2n} + m \) elements, say\n\n\[ X = \left\{ {{x}_{1},\ldots ,{x}_{n},{y}_{1},\ldots ,{y}_{n},{z}_{1},\ldots ,{z}_{m}}\right\} \]\n\nwe shall define an embedding \( \varphi : P \rightarrow \left( {X,{\left\lbrack X\right\rbrack }^{n + 1}, E}\right) \) .\n\nLet us start by setting \( \varphi \left( {a}_{i}\right) \mathrel{\text{:=}} {x}_{i} \) for all \( i = 1,\ldots, n \) . Which \( \left( {n + 1}\right) \) -sets \( Y \subseteq X \) are suitable candidates for the choice of \( \varphi \left( {b}_{i}\right) \) for a given vertex \( {b}_{i} \) ? Clearly those adjacent exactly to the images of the neighbours of \( {b}_{i} \), i.e. those satisfying\n\n\[ Y \cap \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} = \varphi \left( {{N}_{P}\left( {b}_{i}\right) }\right) . \]\n\n(1)\n\nSince \( d\left( {b}_{i}\right) \leq n \), the requirement of (1) leaves at least one of the \( n + 1 \) elements of \( Y \) unspecified. In addition to \( \varphi \left( {{N}_{P}\left( {b}_{i}\right) }\right) \), we may therefore include in each \( Y = \varphi \left( {b}_{i}\right) \) the vertex \( {z}_{i} \) as an ’index’; this ensures that \( \varphi \left( {b}_{i}\right) \neq \varphi \left( {b}_{j}\right) \) for \( i \neq j \), even when \( {b}_{i} \) and \( {b}_{j} \) have the same neighbours in \( P \) . To specify the sets \( Y = \varphi \left( {b}_{i}\right) \) completely, we finally fill them up with ’dummy’ elements \( {y}_{j} \) until \( \left| Y\right| = n + 1 \) . | Yes |
Proposition 9.4.1. For every \( r \in \mathbb{N} \) there is an \( n \in \mathbb{N} \) such that every connected graph of order at least \( n \) contains \( {K}^{r},{K}_{1, r} \) or \( {P}^{r} \) as an induced subgraph. | Proof. Let \( d + 1 \) be the Ramsey number of \( r \), let \( n \mathrel{\text{:=}} \frac{d}{d - 2}{\left( d - 1\right) }^{r} \), and let \( G \) be a graph of order at least \( n \) . If \( G \) has a vertex \( v \) of degree at least \( d + 1 \) then, by Theorem 9.1.1 and the choice of \( d \), either \( N\left( v\right) \) induces a \( {K}^{r} \) in \( G \) or \( \{ v\} \cup N\left( v\right) \) induces a \( {K}_{1, r} \) . On the other hand, if \( \Delta \left( G\right) \leq d \), then by Proposition 1.3.3 \( G \) has radius \( > r \), and hence contains two vertices at a distance \( \geq r \) . Any shortest path in \( G \) between these two vertices contains a \( {P}^{r} \) . | Yes |
Proposition 9.4.2. For every \( r \in \mathbb{N} \) there is an \( n \in \mathbb{N} \) such that every 2-connected graph of order at least \( n \) contains \( {C}^{r} \) or \( {K}_{2, r} \) as a topological minor. | Proof. Let \( d \) be the \( n \) associated with \( r \) in Proposition 9.4.1, and let \( G \) be a 2-connected graph with at least \( \frac{d}{d - 2}{\left( d - 1\right) }^{r} \) vertices. By Proposition 1.3.3, either \( G \) has a vertex of degree \( > d \) or \( \operatorname{diam}G \geq \operatorname{rad}G > r \) . In the latter case let \( a, b \in G \) be two vertices at distance \( > r \) . By Menger’s theorem (3.3.6), \( G \) contains two independent \( a - b \) paths. These form a cycle of length \( > r \) . Assume now that \( G \) has a vertex \( v \) of degree \( > d \) . Since \( G \) is 2- connected, \( G - v \) is connected and thus has a spanning tree; let \( T \) be a minimal tree in \( G - v \) that contains all the neighbours of \( v \) . Then every leaf of \( T \) is a neighbour of \( v \) . By the choice of \( d \), either \( T \) has a vertex of degree \( \geq r \) or \( T \) contains a path of length \( \geq r \), without loss of generality linking two leaves. Together with \( v \), such a path forms a cycle of length \( \geq r \) . A vertex \( u \) of degree \( \geq r \) in \( T \) can be joined to \( v \) by \( r \) independent paths through \( T \), to form a \( T{K}_{2, r} \) . | Yes |
Theorem 9.4.3. (Oporowski, Oxley & Thomas 1993)\n\nFor every \( r \in \mathbb{N} \) there is an \( n \in \mathbb{N} \) such that every 3-connected graph of order at least \( n \) contains a wheel of order \( r \) or a \( {K}_{3, r} \) as a minor. | Null | No |
Theorem 9.4.4. (Oporowski, Oxley & Thomas 1993)\n\nFor every \( r \in \mathbb{N} \) there is an \( n \in \mathbb{N} \) such that every 4-connected graph with at least \( n \) vertices has a minor of order \( \geq r \) that is a double wheel, a crown, a Möbius crown, or a \( {K}_{4, s} \) . | Null | No |
Every graph with \( n \geq 3 \) vertices and minimum degree at least \( n/2 \) has a Hamilton cycle. | Let \( G = \left( {V, E}\right) \) be a graph with \( \left| G\right| = n \geq 3 \) and \( \delta \left( G\right) \geq n/2 \) . Then \( G \) is connected: otherwise, the degree of any vertex in the smallest component \( C \) of \( G \) would be less than \( \left| C\right| \leq n/2 \) .\n\nLet \( P = {x}_{0}\ldots {x}_{k} \) be a longest path in \( G \) . By the maximality of \( P \) , all the neighbours of \( {x}_{0} \) and all the neighbours of \( {x}_{k} \) lie on \( P \) . Hence at least \( n/2 \) of the vertices \( {x}_{0},\ldots ,{x}_{k - 1} \) are adjacent to \( {x}_{k} \), and at least \( n/2 \) of these same \( k < n \) vertices \( {x}_{i} \) are such that \( {x}_{0}{x}_{i + 1} \in E \) . By the pigeon hole principle, there is a vertex \( {x}_{i} \) that has both properties, so we have \( {x}_{0}{x}_{i + 1} \in E \) and \( {x}_{i}{x}_{k} \in E \) for some \( i < k \) (Fig. 10.1.1).\n\nWe claim that the cycle \( C \mathrel{\text{:=}} {x}_{0}{x}_{i + 1}P{x}_{k}{x}_{i}P{x}_{0} \) is a Hamilton cycle of \( G \) . Indeed, since \( G \) is connected, \( C \) would otherwise have a neighbour in \( G - C \), which could be combined with a spanning path of \( C \) into a path longer than \( P \) . | Yes |
Proposition 10.1.2. Every graph \( G \) with \( \left| G\right| \geq 3 \) and \( \alpha \left( G\right) \leq \kappa \left( G\right) \) has a Hamilton cycle. | Proof. Put \( \kappa \left( G\right) = : k \), and let \( C \) be a longest cycle in \( G \). Enumerate the vertices of \( C \) cyclically, say as \( V\left( C\right) = \left\{ {{v}_{i} \mid i \in {\mathbb{Z}}_{n}}\right\} \) with \( {v}_{i}{v}_{i + 1} \in E\left( C\right) \) for all \( i \in {\mathbb{Z}}_{n} \). If \( C \) is not a Hamilton cycle, pick a vertex \( v \in G - C \) and a \( v - C \) fan \( \mathcal{F} = \left\{ {{P}_{i} \mid i \in I}\right\} \) in \( G \), where \( I \subseteq {\mathbb{Z}}_{n} \) and each \( {P}_{i} \) ends in \( {v}_{i} \). Let \( \mathcal{F} \) be chosen with maximum cardinality; then \( v{v}_{j} \notin E\left( G\right) \) for any \( j \notin I \), and \[ \left| \mathcal{F}\right| \geq \min \{ k,\left| C\right| \} \] by Menger's theorem (3.3.4). For every \( i \in I \), we have \( i + 1 \notin I \): otherwise, \( \left( {C \cup {P}_{i} \cup {P}_{i + 1}}\right) - {v}_{i}{v}_{i + 1} \) would be a cycle longer than \( C \) (Fig. 10.1.2, left). Thus \( \left| \mathcal{F}\right| < \left| C\right| \), and hence \( \left| I\right| = \left| \mathcal{F}\right| \geq k \) by (1). Furthermore, \( {v}_{i + 1}{v}_{j + 1} \notin E\left( G\right) \) for all \( i, j \in I \), as otherwise \( \left( {C \cup {P}_{i} \cup {P}_{j}}\right) + {v}_{i + 1}{v}_{j + 1} - {v}_{i}{v}_{i + 1} - {v}_{j}{v}_{j + 1} \) would be a cycle longer than \( C \) (Fig. 10.1.2, right). Hence \( \left\{ {{v}_{i + 1} \mid i \in I}\right\} \cup \{ v\} \) is a set of \( k + 1 \) or more independent vertices in \( G \), contradicting \( \alpha \left( G\right) \leq k \). | Yes |
Theorem 10.1.3. (Tutte 1956)\n\nEvery 4-connected planar graph has a Hamilton cycle. | Null | No |
Corollary 10.2.2. An integer sequence \( \left( {{a}_{1},\ldots ,{a}_{n}}\right) \) such that \( n \geq 2 \) and \( 0 \leq {a}_{1} \leq \ldots \leq {a}_{n} < n \) is path-hamiltonian if and only if every \( i \leq n/2 \) is such that \( {a}_{i} < i \Rightarrow {a}_{n + 1 - i} \geq n - i \) . | Null | No |
If \( G \) is a 2-connected graph, then \( {G}^{2} \) has a Hamilton cycle. | Null | No |
Lemma 10.3.2. Let \( P = {v}_{0}\ldots {v}_{k} \) be a path \( \left( {k \geq 1}\right) \), and let \( G \) be the graph obtained from \( P \) by adding two vertices \( u, w \), together with the edges \( u{v}_{1} \) and \( w{v}_{k} \) (Fig. 10.3.1).\n\n(i) \( {P}^{2} \) contains a path \( Q \) from \( {v}_{0} \) to \( {v}_{1} \) with \( V\left( Q\right) = V\left( P\right) \) and \( {v}_{k - 1}{v}_{k} \in E\left( Q\right) \), such that each of the vertices \( {v}_{1},\ldots ,{v}_{k - 1} \) is bridged by an edge of \( Q \) .\n\n(ii) \( {G}^{2} \) contains disjoint paths \( Q \) from \( {v}_{0} \) to \( {v}_{k} \) and \( {Q}^{\prime } \) from \( u \) to \( w \) , such that \( V\left( Q\right) \cup V\left( {Q}^{\prime }\right) = V\left( G\right) \) and each of the vertices \( {v}_{1},\ldots ,{v}_{k} \) is bridged by an edge of \( Q \) or \( {Q}^{\prime } \) . | Proof. (i) If \( k \) is even, let \( Q \mathrel{\text{:=}} {v}_{0}{v}_{2}\ldots {v}_{k - 2}{v}_{k}{v}_{k - 1}{v}_{k - 3}\ldots {v}_{3}{v}_{1} \) . If \( k \) is odd, let \( Q \mathrel{\text{:=}} {v}_{0}{v}_{2}\ldots {v}_{k - 1}{v}_{k}{v}_{k - 2}\ldots {v}_{3}{v}_{1} \) .\n\n(ii) If \( k \) is even, let \( Q \mathrel{\text{:=}} {v}_{0}{v}_{2}\ldots {v}_{k - 2}{v}_{k} \) ; if \( k \) is odd, let \( Q \mathrel{\text{:=}} \) \( {v}_{0}{v}_{1}{v}_{3}\ldots {v}_{k - 2}{v}_{k} \) . In both cases, let \( {Q}^{\prime } \) be the \( u - w \) path on the remaining vertices of \( {G}^{2} \) . | Yes |
Lemma 10.3.3. Let \( G = \left( {V, E}\right) \) be a cubic multigraph with a Hamilton cycle \( C \) . Let \( e \in E\left( C\right) \) and \( f \in E \smallsetminus E\left( C\right) \) be edges with a common end \( v \) (Fig. 10.3.2). Then there exists a closed walk in \( G \) that traverses \( e \) once, every other edge of \( C \) once or twice, and every edge in \( E \smallsetminus E\left( C\right) \) once. This walk can be chosen to contain the triple \( \left( {e, v, f}\right) \), that is, it traverses \( e \) in the direction of \( v \) and then leaves \( v \) by the edge \( f \) . | Proof. By Proposition 1.2.1, \( C \) has even length. Replace every other edge of \( C \) by a double edge, in such a way that \( e \) does not get replaced. In the arising 4-regular multigraph \( {G}^{\prime } \), split \( v \) into two vertices \( {v}^{\prime },{v}^{\prime \prime } \) , making \( {v}^{\prime } \) incident with \( e \) and \( f \), and \( {v}^{\prime \prime } \) incident with the other two edges at \( v \) (Fig. 10.3.2). By Theorem 1.8.1 this multigraph has an Euler tour, which induces the desired walk in \( G \) . | Yes |
Lemma 10.3.4. For every 2-connected graph \( G \) and \( x \in V\left( G\right) \), there is a cycle \( C \subseteq G \) that contains \( x \) as well as a vertex \( y \neq x \) with \( {N}_{G}\left( y\right) \subseteq V\left( C\right) \) . | Proof. If \( G \) has a Hamilton cycle, there is nothing more to show. If not, let \( {C}^{\prime } \subseteq G \) be any cycle containing \( x \) ; such a cycle exists, since \( G \) is 2-connected. Let \( D \) be a component of \( G - {C}^{\prime } \) . Assume that \( {C}^{\prime } \) and \( D \) are chosen so that \( \left| D\right| \) is minimal. Since \( G \) is 2-connected, \( D \) has at least two neighbours on \( {C}^{\prime } \) . Then \( {C}^{\prime } \) contains a path \( P \) between two such neighbours \( u \) and \( v \), whose interior \( \overset{ \circ }{P} \) does not contain \( x \) and has no neighbour in \( D \) (Fig. 10.3.3). Replacing \( P \) in \( {C}^{\prime } \) by a \( u - v \) path through \( D \), we obtain a cycle \( C \) that contains \( x \) and a vertex \( y \in D \) . If \( y \) had a neighbour \( z \) in \( G - C \), then \( z \) would lie in a component \( {D}^{\prime } \subsetneqq D \) of \( G - C \), contradicting the choice of \( {C}^{\prime } \) and \( D \) . Hence all the neighbours of \( y \) lie on \( C \), and \( C \) satisfies the assertion of the lemma. | Yes |
Proposition 11.1.1. The events \( {A}_{e} \) are independent and occur with probability \( p \) . | Proof. By definition,\n\n\[ \n{A}_{e} = \left\{ {1}_{e}\right\} \times \mathop{\prod }\limits_{{{e}^{\prime } \neq e}}{\Omega }_{{e}^{\prime }} \n\] \n\nSince \( P \) is the product measure of all the measures \( {P}_{e} \), this implies\n\n\[ \nP\left( {A}_{e}\right) = p \cdot \mathop{\prod }\limits_{{{e}^{\prime } \neq e}}1 = p. \n\] \n\nSimilarly, if \( \left\{ {{e}_{1},\ldots ,{e}_{k}}\right\} \) is any subset of \( {\left\lbrack V\right\rbrack }^{2} \), then\n\n\[ \n\begin{aligned} P\left( {{A}_{{e}_{1}} \cap \ldots \cap {A}_{{e}_{k}}}\right) & = P\left( {\left\{ {1}_{{e}_{1}}\right\} \times \ldots \times \left\{ {1}_{{e}_{k}}\right\} \times \mathop{\prod }\limits_{{e \notin \left\{ {{e}_{1},\ldots ,{e}_{k}}\right\} }}{\Omega }_{e}}\right) \\ & = {p}^{k} \end{aligned} \n\] \n\n\[ \n= P\left( {A}_{{e}_{1}}\right) \cdots P\left( {A}_{{e}_{k}}\right) . \n\] \n\nAs noted before, \( P \) is determined uniquely by the value of \( p \) and our assumption that the events \( {A}_{e} \) are independent. In order to calculate probabilities in \( \mathcal{G}\left( {n, p}\right) \), it therefore generally suffices to work with these two assumptions: our concrete model for \( \mathcal{G}\left( {n, p}\right) \) has served its purpose and will not be needed again. | Yes |
For all integers \( n, k \) with \( n \geq k \geq 2 \), the probability that \( G \in \mathcal{G}\left( {n, p}\right) \) has a set of \( k \) independent vertices is at most | The probability that a fixed \( k \) -set \( U \subseteq V \) is independent in \( G \) is \( {q}^{\left( \begin{matrix} k \\ 2 \end{matrix}\right) } \) . The assertion thus follows from the fact that there are only \( \left( \begin{array}{l} n \\ k \end{array}\right) \) such sets \( U \) . | No |
For every integer \( k \geq 3 \), the Ramsey number of \( k \) satisfies\n\n\[ R\left( k\right) > {2}^{k/2}\text{.} \] | Proof. For \( k = 3 \) we trivially have \( R\left( 3\right) \geq 3 > {2}^{3/2} \), so let \( k \geq 4 \) . We show that, for all \( n \leq {2}^{k/2} \) and \( G \in \mathcal{G}\left( {n,\frac{1}{2}}\right) \), the probabilities \( P\left\lbrack {\alpha \left( G\right) \geq k}\right\rbrack \) and \( P\left\lbrack {\omega \left( G\right) \geq k}\right\rbrack \) are both less than \( \frac{1}{2} \) .\n\nSince \( p = q = \frac{1}{2} \), Lemma 11.1.2 and the analogous assertion for \( \omega \left( G\right) \) imply the following for all \( n \leq {2}^{k/2} \) (use that \( k! > {2}^{k} \) for \( k \geq 4 \) ):\n\n\[ P\left\lbrack {\alpha \left( G\right) \geq k}\right\rbrack, P\left\lbrack {\omega \left( G\right) \geq k}\right\rbrack \leq \left( \begin{array}{l} n \\ k \end{array}\right) {\left( \frac{1}{2}\right) }^{\left( \begin{matrix} k \\ 2 \end{matrix}\right) }\n\n\[ < \left( {{n}^{k}/{2}^{k}}\right) {2}^{-\frac{1}{2}k\left( {k - 1}\right) }\n\n\[ \leq \left( {{2}^{{k}^{2}/2}/{2}^{k}}\right) {2}^{-\frac{1}{2}k\left( {k - 1}\right) }\n\n\[ = {2}^{-k/2}\n\n\[ < \frac{1}{2}\text{.} \] | Yes |
Lemma 11.1.4. (Markov's Inequality)\n\nLet \( X \geq 0 \) be a random variable on \( \mathcal{G}\left( {n, p}\right) \) and \( a > 0 \) . Then\n\n\[ P\left\lbrack {X \geq a}\right\rbrack \leq E\left( X\right) /a. \] | Proof.\n\n\[ E\left( X\right) = \mathop{\sum }\limits_{{G \in \mathcal{G}\left( {n, p}\right) }}P\left( {\{ G\} }\right) \cdot X\left( G\right) \]\n\n\[ \geq \mathop{\sum }\limits_{\substack{{G \in \mathcal{G}\left( {n, p}\right) } \\ {X\left( G\right) \geq a} }}P\left( {\{ G\} }\right) \cdot X\left( G\right) \]\n\n\[ \geq \mathop{\sum }\limits_{\substack{{G \in \mathcal{G}\left( {n, p}\right) } \\ {X\left( G\right) \geq a} }}P\left( {\{ G\} }\right) \cdot a \]\n\n\[ = P\left\lbrack {X \geq a}\right\rbrack \cdot a. \] | Yes |
Lemma 11.1.5. The expected number of \( k \) -cycles in \( G \in \mathcal{G}\left( {n, p}\right) \) is\n\n\[ E\left( X\right) = \frac{{\left( n\right) }_{k}}{2k}{p}^{k} \] | Proof. For every \( k \) -cycle \( C \) with vertices in \( V = \{ 0,\ldots, n - 1\} \), the vertex set of the graphs in \( \mathcal{G}\left( {n, p}\right) \), let \( {X}_{C} : \mathcal{G}\left( {n, p}\right) \rightarrow \{ 0,1\} \) denote the indicator random variable of \( C \) :\n\n\[ {X}_{C} : G \mapsto \left\{ \begin{array}{ll} 1 & \text{ if }C \subseteq G \\ 0 & \text{ otherwise. } \end{array}\right. \]\n\nSince \( {X}_{C} \) takes only 1 as a positive value, its expectation \( E\left( {X}_{C}\right) \) equals the measure \( P\left\lbrack {{X}_{C} = 1}\right\rbrack \) of the set of all graphs in \( \mathcal{G}\left( {n, p}\right) \) that contain \( C \) . But this is just the probability that \( C \subseteq G \) :\n\n\[ E\left( {X}_{C}\right) = P\left\lbrack {C \subseteq G}\right\rbrack = {p}^{k}. \]\n\n(1)\n\nHow many such cycles \( C = {v}_{0}\ldots {v}_{k - 1}{v}_{0} \) are there? There are \( {\left( n\right) }_{k} \) sequences \( {v}_{0}\ldots {v}_{k - 1} \) of distinct vertices in \( V \), and each cycle is identified by \( {2k} \) of those sequences - so there are exactly \( {\left( n\right) }_{k}/{2k} \) such cycles.\n\nOur random variable \( X \) assigns to every graph \( G \) its number of \( k \) - cycles. Clearly, this is the sum of all the values \( {X}_{C}\left( G\right) \), where \( C \) varies over the \( {\left( n\right) }_{k}/{2k} \) cycles of length \( k \) with vertices in \( V \) :\n\n\[ X = \mathop{\sum }\limits_{C}{X}_{C} \]\n\nSince the expectation is linear, (1) thus implies\n\n\[ E\left( X\right) = E\left( {\mathop{\sum }\limits_{C}{X}_{C}}\right) = \mathop{\sum }\limits_{C}E\left( {X}_{C}\right) = \frac{{\left( n\right) }_{k}}{2k}{p}^{k} \]\n\nas claimed. | Yes |
Lemma 11.2.1. Let \( k > 0 \) be an integer, and let \( p = p\left( n\right) \) be a function of \( n \) such that \( p \geq \left( {{6k}\ln n}\right) {n}^{-1} \) for \( n \) large. Then\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}P\left\lbrack {\alpha \geq \frac{1}{2}n/k}\right\rbrack = 0. \] | Proof. For all integers \( n, r \) with \( n \geq r \geq 2 \), and all \( G \in \mathcal{G}\left( {n, p}\right) \), Lemma 11.1.2 implies\n\n\[ P\left\lbrack {\alpha \geq r}\right\rbrack \leq \left( \begin{array}{l} n \\ r \end{array}\right) {q}^{\left( \begin{array}{l} r \\ 2 \end{array}\right) }\n\]\n\[ \leq {n}^{r}{q}^{\left( \begin{array}{l} r \\ 2 \end{array}\right) }\n\]\n\[ = {\left( n{q}^{\left( {r - 1}\right) /2}\right) }^{r}\n\]\n\[ \leq {\left( n{e}^{-p\left( {r - 1}\right) /2}\right) }^{r}\n\]\nhere, the last inequality follows from the fact that \( 1 - p \leq {e}^{-p} \) for all \( p \) . (Compare the functions \( x \mapsto {e}^{x} \) and \( x \mapsto x + 1 \) for \( x = - p \) .) Now if \( p \geq \left( {{6k}\ln n}\right) {n}^{-1} \) and \( r \geq \frac{1}{2}n/k \), then the term under the exponent satisfies\n\n\[ n{e}^{-p\left( {r - 1}\right) /2} = n{e}^{-{pr}/2 + p/2}\n\]\n\[ \leq n{e}^{-\left( {3/2}\right) \ln n + p/2}\n\]\n\[ \leq n{n}^{-3/2}{e}^{1/2}\n\]\n\[ = \sqrt{e}/\sqrt{n}\underset{n \rightarrow \infty }{ \rightarrow }0\n\]\n\nSince \( p \geq \left( {{6k}\ln n}\right) {n}^{-1} \) for \( n \) large, we thus obtain for \( r \mathrel{\text{:=}} \left\lceil {\frac{1}{2}n/k}\right\rceil \)\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}P\left\lbrack {\alpha \geq \frac{1}{2}n/k}\right\rbrack = \mathop{\lim }\limits_{{n \rightarrow \infty }}P\left\lbrack {\alpha \geq r}\right\rbrack = 0,\n\]\n\nas claimed. | Yes |
For every integer \( k \) there exists a graph \( H \) with girth \( g\left( H\right) > k \) and chromatic number \( \chi \left( H\right) > k \) . | Proof. Assume that \( k \geq 3 \), fix \( \epsilon \) with \( 0 < \epsilon < 1/k \), and let \( p \mathrel{\text{:=}} {n}^{\epsilon - 1} \) . Let\n\n\( X\left( G\right) \) denote the number of short cycles in a random graph \( G \in \mathcal{G}\left( {n, p}\right) \) ,\n\n\( p,\epsilon, X \)\n\ni.e. its number of cycles of length at most \( k \) .\n\nBy Lemma 11.1.5, we have\n\n\[ E\left( X\right) = \mathop{\sum }\limits_{{i = 3}}^{k}\frac{{\left( n\right) }_{i}}{2i}{p}^{i} \leq \frac{1}{2}\mathop{\sum }\limits_{{i = 3}}^{k}{n}^{i}{p}^{i} \leq \frac{1}{2}\left( {k - 2}\right) {n}^{k}{p}^{k}; \]\n\nnote that \( {\left( np\right) }^{i} \leq {\left( np\right) }^{k} \), because \( {np} = {n}^{\epsilon } \geq 1 \) . By Lemma 11.1.4,\n\n\[ P\left\lbrack {X \geq n/2}\right\rbrack \leq E\left( X\right) /\left( {n/2}\right) \]\n\n\[ \leq \left( {k - 2}\right) {n}^{k - 1}{p}^{k} \]\n\n\[ = \left( {k - 2}\right) {n}^{k - 1}{n}^{\left( {\epsilon - 1}\right) k} \]\n\n\[ = \left( {k - 2}\right) {n}^{{k\epsilon } - 1}. \]\n\nAs \( {k\epsilon } - 1 < 0 \) by our choice of \( \epsilon \), this implies that\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}P\left\lbrack {X \geq n/2}\right\rbrack = 0. \]\n\nLet \( n \) be large enough that \( P\left\lbrack {X \geq n/2}\right\rbrack < \frac{1}{2} \) and \( P\left\lbrack {\alpha \geq \frac{1}{2}n/k}\right\rbrack < \frac{1}{2} \) ; the latter is possible by our choice of \( p \) and Lemma 11.2.1. Then there is a graph \( G \in \mathcal{G}\left( {n, p}\right) \) with fewer than \( n/2 \) short cycles and \( \alpha \left( G\right) < \) \( \frac{1}{2}n/k \) . From each of those cycles delete a vertex, and let \( H \) be the graph obtained. Then \( \left| H\right| \geq n/2 \) and \( H \) has no short cycles, so \( g\left( H\right) > k \) . By definition of \( G \),\n\n\[ \chi \left( H\right) \geq \frac{\left| H\right| }{\alpha \left( H\right) } \geq \frac{n/2}{\alpha \left( G\right) } > k. \] | Yes |
Corollary 11.2.3. There are graphs with arbitrarily large girth and arbitrarily large values of the invariants \( \kappa ,\varepsilon \) and \( \delta \) . | Proof. Apply Corollary 5.2.3 and Theorem 1.4.3. | No |
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