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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 ... | Yes |
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 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 |
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, thi... | 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... | 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 |
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( {\ma... | 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 \( H \) -f... | 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 ... | 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 |
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... | 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 |
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 \( \varepsilon \left( H\right) > \varepsilon \left( G\right) - {r}^{2} \geq 4{r}^{2} \).... | 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... | Yes |
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... | 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... | Yes |
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 |
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 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 8.4.4. 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 \( A - {B}^{\prime } \) 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... | Yes |
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... | Yes |
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| \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... | 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}}\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 |
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}_... | 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 |
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 \(... | 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 \( U \) . | Yes |
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 |
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