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Proposition 4.2.7. The face boundaries in a 3-connected plane graph are precisely its non-separating induced cycles. | Proof. Let \( G \) be a 3-connected plane graph, and let \( C \subseteq G \) . If \( C \) is a (4.1.1) non-separating induced cycle, then by the Jordan curve theorem its two\n\n(4.1.2)\n\nfaces cannot both contain points of \( G \smallsetminus C \) . Therefore it bounds a face of \( G \) .\n\nConversely, suppose that \... | Yes |
Theorem 4.2.9. (Euler's Formula)\n\nLet \( G \) be a connected plane graph with \( n \) vertices, \( m \) edges, and \( \ell \) faces. Then\n\n\[ n - m + \ell = 2. \] | Proof. We fix \( n \) and apply induction on \( m \) . For \( m \leq n - 1, G \) is a tree\n\n(1.5.1)\n\n\( \left( {1.5.3}\right) \) and \( m = n - 1 \) (why?), so the assertion follows from Proposition 4.2.4.\n\nNow let \( m \geq n \) . Then \( G \) has an edge \( e \) that lies on a cycle; let \( {G}^{\prime } \mathr... | Yes |
Corollary 4.2.10. A plane graph with \( n \geq 3 \) vertices has at most \( {3n} - 6 \) edges. Every plane triangulation with \( n \) vertices has \( {3n} - 6 \) edges. | Proof. By Proposition 4.2.8 it suffices to prove the second assertion. In a plane triangulation \( G \), every face boundary contains exactly three edges, and every edge lies on the boundary of exactly two faces (Lemma 4.2.2). The bipartite graph on \( E\left( G\right) \cup F\left( G\right) \) with edge set \( \{ {ef} ... | Yes |
Any two planar embeddings of a 3-connected graph are equivalent. | Proof. Let \( G \) be a 3-connected graph with planar embeddings \( \rho : G \rightarrow H \) and \( {\rho }^{\prime } : G \rightarrow {H}^{\prime } \) . By Theorem 4.3.1 it suffices to show that \( {\rho }^{\prime } \circ {\rho }^{-1} \) is a graph-theoretical isomorphism, i.e. that \( \rho \left( C\right) \) bounds a... | No |
Lemma 4.4.2. A graph contains \( {K}^{5} \) or \( {K}_{3,3} \) as a minor if and only if it contains \( {K}^{5} \) or \( {K}_{3,3} \) as a topological minor. | Proof. By Proposition 1.7.2 it suffices to show that every graph \( G \) with a \( {K}^{5} \) minor contains either \( {K}^{5} \) as a topological minor or \( {K}_{3,3} \) as a minor. So suppose that \( G \succcurlyeq {K}^{5} \), and let \( K \subseteq G \) be minimal such that \( K = M{K}^{5} \). Then every branch set... | Yes |
Lemma 4.4.4. Let \( \mathcal{X} \) be a set of 3-connected graphs. Let \( G \) be a graph with \( \kappa \left( G\right) \leq 2 \), and let \( {G}_{1},{G}_{2} \) be proper induced subgraphs of \( G \) such that \( G = {G}_{1} \cup {G}_{2} \) and \( \left| {{G}_{1} \cap {G}_{2}}\right| = \kappa \left( G\right) \) . If \... | Proof. Note first that every vertex \( v \in S \mathrel{\text{:=}} V\left( {{G}_{1} \cap {G}_{2}}\right) \) has a neighbour in every component of \( {G}_{i} - S, i = 1,2 \) : otherwise \( S \smallsetminus \{ v\} \) would separate \( G \), contradicting \( \left| S\right| = \kappa \left( G\right) \) . By the maximality ... | Yes |
Lemma 4.4.5. If \( \left| G\right| \geq 4 \) and \( G \) is edge-maximal with \( T{K}^{5}, T{K}_{3,3} \nsubseteq G \) , then \( G \) is 3-connected. | Proof. We apply induction on \( \left| G\right| \) . For \( \left| G\right| = 4 \), we have \( G = {K}^{4} \) and the assertion holds. Now let \( \left| G\right| > 4 \), and let \( G \) be edge-maximal without a \( T{K}^{5} \) or \( T{K}_{3,3} \) . Suppose \( \kappa \left( G\right) \leq 2 \), and choose \( {G}_{1} \) a... | Yes |
Theorem 4.4.6. (Kuratowski 1930; Wagner 1937) \( \left\lbrack \begin{array}{l} {4.5.1} \\ {12.4.3} \end{array}\right\rbrack \)\n\nThe following assertions are equivalent for graphs \( G \) :\n\n(i) \( G \) is planar;\n\n(ii) \( G \) contains neither \( {K}^{5} \) nor \( {K}_{3,3} \) as a minor;\n\n(iii) \( G \) contain... | Proof. Combine Corollary 4.2.11 with Lemmas 4.4.2, 4.4.3 and 4.4.5. | No |
Corollary 4.4.7. Every maximal planar graph with at least four vertices is 3-connected. | Proof. Apply Lemma 4.4.5 and Theorem 4.4.6. | No |
A 3-connected graph is planar if and only if every edge lies on at most (equivalently: exactly) two non-separating induced cycles. | The forward implication follows from Propositions 4.2.7 and (4.2.6) (4.2.7) 4.2.2 (and Proposition 4.2.6 for the 'exactly two' version); the backward implication follows from Theorems 3.2.3 and 4.5.1. | Yes |
Proposition 4.6.1. For any connected plane multigraph \( G \), an edge set \( E \subseteq E\left( G\right) \) is the edge set of a cycle in \( G \) if and only if \( {E}^{ * } \mathrel{\text{:=}} \left\{ {{e}^{ * } \mid e \in E}\right\} \) is a minimal cut in \( {G}^{ * } \) . | Proof. By conditions (i) and (ii) in the definition of \( {G}^{ * } \), two vertices \( {v}^{ * }\left( {f}_{1}\right) \) and \( {v}^{ * }\left( {f}_{2}\right) \) of \( {G}^{ * } \) lie in the same component of \( {G}^{ * } - {E}^{ * } \) if and only if \( {f}_{1} \) and \( {f}_{2} \) lie in the same region of \( {\mat... | Yes |
Proposition 4.6.2. If \( {G}^{ * } \) is an abstract dual of \( G \), then the cut space of \( {G}^{ * } \) is the cycle space of \( G \), i.e., \[ {\mathcal{C}}^{ * }\left( {G}^{ * }\right) = \mathcal{C}\left( G\right) \] | Proof. Since the cycles of \( G \) are precisely the bonds of \( {G}^{ * } \), the subspace \( \mathcal{C}\left( G\right) \) they generate in \( \mathcal{E}\left( G\right) = \mathcal{E}\left( {G}^{ * }\right) \) is the same as the subspace generated by the bonds in \( {G}^{ * } \) . By Lemma 1.9.4, \( {}^{5} \) this is... | No |
A graph is planar if and only if it has an abstract dual. | Proof. Let \( G \) be a planar graph, and consider any drawing. Every \( {\text{component}}^{6}C \) of this drawing has a plane dual \( {C}^{ * } \) . Consider these \( {C}^{ * } \) as abstract multigraphs, and let \( {G}^{ * } \) be their disjoint union. Then the bonds of \( {G}^{ * } \) are precisely the minimal cuts... | 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 ... | 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 n... | 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 \( {xy} \in G \), an... | 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 colou... | 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 \( v \) w... | 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 \( ... | Yes |
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, thi... | Yes |
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... | 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... | 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 |
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( {\ma... | Yes |
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 \( H \) -f... | Yes |
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 ... | 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},... | 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}... | Yes |
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... | 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... | 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 hy... | 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}... | 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\n... | Yes |
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 \( \varepsilon \left( H\right) > \varepsilon \left( G\right) - {r}^{2} \geq 4{r}^{2} \).... | 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 \( x \in X \)... | 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... | 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 wi... | 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 \( \left| G\right| \) t... | 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 7.3.8. (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... | 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... | 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 - \epsil... | 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\... | 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... | 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 \( \{ ... | 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... | 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, ... | 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 \( T \) separate... | 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 t... | 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... | 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 \( {U}_{1},{W}_{1} \) an... | Yes |
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 e... | 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\paral... | 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 i... | Yes |
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 \( A - {B}^{\prime } \) separa... | 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 cy... | 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 ma... | 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| ... | 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... | 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 ... | 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... | 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 8.5.4. 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| \small... | 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... | 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... | 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 ... | 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 gene... | 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 leas... | 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\nLe... | 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 edge... | 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 se... | 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.... | 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 pr... | 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}_... | 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 \( {\le... | 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) \lef... | 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... | 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 ... | 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 \righ... | No |
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}}\righ... | 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\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 \( \operatorname{diam}G \geq \operatorname{rad}G > r \) ... | Yes |
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}_... | 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 \(... | Yes |
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 \(... | 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{\... | 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 ... | 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 \( {v}^{\prime } \) inc... | 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 \... | 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 } \n... | 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 {\ome... | 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\[ ... | 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\{ \be... | 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... | 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 cycle... | 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 |
Proposition 11.3.1. For every constant \( p \in \left( {0,1}\right) \) and every graph \( H \) , almost every \( G \in \mathcal{G}\left( {n, p}\right) \) contains an induced copy of \( H \) . | Proof. Let \( H \) be given, and \( k \mathrel{\text{:=}} \left| H\right| \) . If \( n \geq k \) and \( U \subseteq \{ 0,\ldots, n - 1\} \) is a fixed set of \( k \) vertices of \( G \), then \( G\left\lbrack U\right\rbrack \) is isomorphic to \( H \) with a certain probability \( r > 0 \) . This probability \( r \) de... | Yes |
Lemma 11.3.2. For every constant \( p \in \left( {0,1}\right) \) and \( i, j \in \mathbb{N} \), almost every graph \( G \in \mathcal{G}\left( {n, p}\right) \) has the property \( {\mathcal{P}}_{i, j} \) . | Proof. For fixed \( U, W \) and \( v \in G - \left( {U \cup W}\right) \), the probability that \( v \) is adjacent to all the vertices in \( U \) but to none in \( W \), is\n\n\[ \n{p}^{\left| U\right| }{q}^{\left| W\right| } \geq {p}^{i}{q}^{j} \n\]\n\nHence, the probability that no suitable \( v \) exists for these \... | Yes |
Corollary 11.3.3. For every constant \( p \in \left( {0,1}\right) \) and \( k \in \mathbb{N} \), almost every graph in \( \mathcal{G}\left( {n, p}\right) \) is \( k \) -connected. | Proof. By Lemma 11.3.2, it is enough to show that every graph in \( {\mathcal{P}}_{2, k - 1} \) is \( k \) -connected. But this is easy: any graph in \( {\mathcal{P}}_{2, k - 1} \) has order at least \( k + 2 \), and if \( W \) is a set of fewer than \( k \) vertices, then by definition of \( {\mathcal{P}}_{2, k - 1} \... | No |
Proposition 11.3.4. For every constant \( p \in \left( {0,1}\right) \) and every \( \epsilon > 0 \) , almost every graph \( G \in \mathcal{G}\left( {n, p}\right) \) has chromatic number\n\n\[ \chi \left( G\right) > \frac{\log \left( {1/q}\right) }{2 + \epsilon } \cdot \frac{n}{\log n}. \] | Proof. For any fixed \( n \geq k \geq 2 \), Lemma 11.1.2 implies\n\n\[ P\left\lbrack {\alpha \geq k}\right\rbrack \leq \left( \begin{array}{l} n \\ k \end{array}\right) {q}^{\left( \begin{array}{l} k \\ 2 \end{array}\right) }\n\n\[ \leq {n}^{k}{q}^{\left( \begin{matrix} k \\ 2 \end{matrix}\right) }\n\n\[ = {q}^{k\frac{... | Yes |
Theorem 11.3.5. (Erdős and Rényi 1963)\n\nWith probability 1, a random graph \( G \in \mathcal{G}\left( {{\aleph }_{0}, p}\right) \) with \( 0 < p < 1 \) is isomorphic to the Rado graph \( R \) . | Proof. Given fixed disjoint finite sets \( U, W \subseteq \mathbb{N} \), the probability that a vertex \( v \notin U \cup W \) is not joined to \( U \cup W \) as expressed in property \( \left( *\right) \) of Chapter 8.3 (i.e., is not joined to all of \( U \) or is joined to some vertex in \( W \) ) is some number \( r... | Yes |
Lemma 11.4.1. (Chebyshev's Inequality)\n\nFor all real \( \lambda > 0 \) ,\n\n\[ P\left\lbrack {\left| {X - \mu }\right| \geq \lambda }\right\rbrack \leq {\sigma }^{2}/{\lambda }^{2} \]\n\n\( \left( {11.1.4}\right) \) | Proof. By Lemma 11.1.4 and definition of \( {\sigma }^{2} \) ,\n\n\[ P\left\lbrack {\left| {X - \mu }\right| \geq \lambda }\right\rbrack = P\left\lbrack {{\left( X - \mu \right) }^{2} \geq {\lambda }^{2}}\right\rbrack \leq {\sigma }^{2}/{\lambda }^{2}. \] | Yes |
Lemma 11.4.2. If \( \mu > 0 \) for \( n \) large, and \( {\sigma }^{2}/{\mu }^{2} \rightarrow 0 \) as \( n \rightarrow \infty \), then \( X\left( G\right) > 0 \) for almost all \( G \in \mathcal{G}\left( {n, p}\right) \) . | Proof. Any graph \( G \) with \( X\left( G\right) = 0 \) satisfies \( \left| {X\left( G\right) - \mu }\right| = \mu \) . Hence Lemma 11.4.1 implies with \( \lambda \mathrel{\text{:=}} \mu \) that\n\n\[ P\left\lbrack {X = 0}\right\rbrack \leq P\left\lbrack {\left| {X - \mu }\right| \geq \mu }\right\rbrack \leq {\sigma }... | Yes |
Corollary 11.4.6. If \( k \geq 2 \), then \( t\left( n\right) = {n}^{-2/\left( {k - 1}\right) } \) is a threshold function for the property of containing a \( {K}^{k} \) . | Proof. \( {K}^{k} \) is balanced, because \( \varepsilon \left( {K}^{i}\right) = \frac{1}{2}\left( {i - 1}\right) < \frac{1}{2}\left( {k - 1}\right) = \varepsilon \left( {K}^{k}\right) \) for \( i < k \) . With \( \ell \mathrel{\text{:=}} \begin{Vmatrix}{K}^{k}\end{Vmatrix} = \frac{1}{2}k\left( {k - 1}\right) \), we ob... | Yes |
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