Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
Theorem 10.35 Every 3-connected nonplanar graph has a Kuratowski minor. | Proof Let \( G \) be a 3-connected nonplanar graph. We may assume that \( G \) is simple. Because all graphs on four or fewer vertices are planar, we have \( n \geq 5 \) . We proceed by induction on \( n \) . By Theorem 9.10, \( G \) contains an edge \( e = {xy} \) such that \( H \mathrel{\text{:=}} \) \( G/e \) is 3-c... | Yes |
The tree-width of any graph is equal to its bramble number. | For a beautiful, unified proof of this and other related duality theorems, we refer the reader to the article by Amini et al. (2007). | No |
Theorem 11.2 The Four-Colour Theorem\n\nEvery plane graph without cut edges is 4-face-colourable. | More recently, a somewhat simpler (but still complicated) proof of this theorem, based on the same general approach, was obtained by Robertson et al. (1997a). | No |
A 3-connected cubic plane graph is 4-face-colourable if and only if it is 3-edge-colourable. | Let \( G \) be a 3-connected cubic plane graph. First, suppose that that \( G \) has a proper 4-face-colouring. It is of course immaterial which symbols are used as the 'colours'. For mathematical convenience, we denote them by the vectors \( {\alpha }_{0} = \left( {0,0}\right) ,{\alpha }_{1} = \left( {1,0}\right) ,{\a... | Yes |
Theorem 11.6 The Five Colour Theorem\n\nEvery loopless planar graph is 5-colourable. | Proof By induction on the number of vertices. As observed in our earlier discussion, it suffices to prove the theorem for 3-connected triangulations. So let \( G \) be such a triangulation. By Corollary 10.22, \( G \) has a vertex \( v \) of degree at most five. Consider the plane graph \( H \mathrel{\text{:=}} G - v \... | Yes |
Given a positive integer \( k \), what is the greatest number of sequences of signals (or words) of length \( k \) that can be transmitted with no possibility of confusion at the receiving end? | To translate this problem into graph theory, we need the concept of the strong product of two graphs \( G \) and \( H \) . This is the graph \( G \boxtimes H \) whose vertex set is \( V\left( G\right) \times V\left( H\right) \), vertices \( \left( {u, x}\right) \) and \( \left( {v, y}\right) \) being adjacent if and on... | Yes |
Let \( D \) be a digraph which contains no directed odd cycle. Then \( D \) has a kernel. | Proof By induction on \( n \) . If \( D \) is strong, then \( D \) is bipartite (Exercise 3.4.11b) and each class of the bipartition is a kernel of \( D \) . If \( D \) is not strong, let \( {D}_{1} \) be a minimal strong component of \( D \) (one that dominates no other strong component; see Exercise 3.4.6), and set \... | No |
Theorem 12.5 For any two integers \( k \geq 2 \) and \( \ell \geq 2 \) ,\n\n\[ r\left( {k,\ell }\right) \leq r\left( {k,\ell - 1}\right) + r\left( {k - 1,\ell }\right) \]\n\nFurthermore, if \( r\left( {k,\ell - 1}\right) \) and \( r\left( {k - 1,\ell }\right) \) are both even, strict inequality holds in (12.5). | Proof Let \( G \) be a graph on \( r\left( {k,\ell - 1}\right) + r\left( {k - 1,\ell }\right) \) vertices, and let \( v \in V \) . We distinguish two cases:\n\n1. Vertex \( v \) is nonadjacent to a set \( S \) of at least \( r\left( {k,\ell - 1}\right) \) vertices.\n\n2. Vertex \( v \) is adjacent to a set \( T \) of a... | Yes |
Theorem 12.6 For all positive integers \( k \) and \( \ell \) ,\n\n\[ r\left( {k,\ell }\right) \leq \left( \begin{matrix} k + \ell - 2 \\ k - 1 \end{matrix}\right) \] | Proof By induction on \( k + \ell \) . Using (12.3) and (12.4), we see that the theorem holds when \( k + \ell \leq 5 \) . Let \( m \) and \( n \) be positive integers, and assume that the theorem is valid for all positive integers \( k \) and \( \ell \) such that \( 5 \leq k + \ell < m + n \) . Then, by Theorem 12.5 a... | Yes |
Theorem 12.8 For all positive integers \( k \) ,\n\n\[ r\left( {k, k}\right) \geq {2}^{k/2} \] | Proof Because \( r\left( {1,1}\right) = 1 \) and \( r\left( {2,2}\right) = 2 \), we may assume that \( k \geq 3 \) . As in Section 1.2, we denote by \( {\mathcal{G}}_{n} \) the set of simple graphs with vertex set \( \left\{ {{v}_{1},{v}_{2},\ldots ,{v}_{n}}\right\} \) . Let \( {\mathcal{G}}_{n}^{k} \) be the set of th... | Yes |
Theorem 12.9 For all positive integers \( {t}_{i},1 \leq i \leq k \) , | \[ r\left( {{t}_{1},{t}_{2},\ldots ,{t}_{k}}\right) \leq r\left( {{t}_{1} - 1,{t}_{2},\ldots ,{t}_{k}}\right) + r\left( {{t}_{1},{t}_{2} - 1,\ldots ,{t}_{k}}\right) + \cdots + r\left( {{t}_{1},{t}_{2},\ldots ,{t}_{k} - 1}\right) - k + 2 \] | Yes |
Theorem 12.11 SCHUR'S THEOREM\n\nLet \( \\left\\{ {{A}_{1},{A}_{2},\\ldots ,{A}_{n}}\\right\\} \) be a partition of the set of integers \( \\left\\{ {1,2,\\ldots ,{r}_{n}}\\right\\} \) into \( n \) subsets. Then some \( {A}_{i} \) contains three integers \( x, y \), and \( z \) satisfying the equation \( x + y = z \) . | Proof Consider the complete graph whose vertex set is \( \\left\\{ {1,2,\\ldots ,{r}_{n}}\\right\\} \). Colour the edges of this graph in colours \( 1,2,\\ldots, n \) by the rule that the edge \( {uv} \) is assigned colour \( i \) if \( \\left| {u - v}\\right| \\in {A}_{i} \). By Ramsey’s Theorem (12.9), there exists a... | Yes |
Lemma 12.13 Let \( \left( {X, Y}\right) \) be a regular pair of density \( d \), let \( {Y}^{\prime } \) be a large subset of \( Y \), and let \( S \) be the set of vertices of \( X \) which have fewer than \( \left( {d - \epsilon }\right) \left| {Y}^{\prime }\right| \) neighbours in \( {Y}^{\prime } \) . Then \( S \) ... | Proof Consider a large subset \( {X}^{\prime } \) of \( X \) . Because \( \left( {X, Y}\right) \) is a regular pair, \( d\left( {{X}^{\prime },{Y}^{\prime }}\right) \geq d - \epsilon \) . Hence \( e\left( {{X}^{\prime },{Y}^{\prime }}\right) \geq \left( {d - \epsilon }\right) \left| {X}^{\prime }\right| \left| {Y}^{\pr... | Yes |
Theorem 12.15 For any simple graph \( F \)\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{\operatorname{ex}\left( {n, F}\right) }{{n}^{2}} = \frac{1}{2}\left( \frac{\chi \left( F\right) - 2}{\chi \left( F\right) - 1}\right) \] | Proof Let \( k = \chi \left( F\right) \) and let \( t \) be the largest size of a colour class in a proper \( k \) -colouring of \( F \) . Then \( F \subseteq {T}_{k,{tk}} \) . Therefore, by the Erdős-Stone Theorem, for any \( d > 0 \), there is an integer \( q \), depending on \( k, t \), and \( d \), such that every ... | Yes |
Proposition 12.17 Let \( G \) be a graph, let \( \mathcal{P} \) be a family of disjoint subsets of \( V \) , and let \( \mathcal{Q} \) be a refinement of \( \mathcal{P} \) . Then \( \rho \left( \mathcal{Q}\right) \geq \rho \left( \mathcal{P}\right) \) . | Proof It suffices to show that the conclusion holds when \( \mathcal{Q} \) is obtained from \( \mathcal{P} \) by partitioning one set \( X \in \mathcal{P} \) into two nonempty sets \( {X}_{1},{X}_{2} \) . We have:\n\n\[ \rho \left( \mathcal{Q}\right) - \rho \left( \mathcal{P}\right) = \rho \left( {{X}_{1},{X}_{2}}\righ... | Yes |
Lemma 12.18 Let \( \\left( {X, Y}\\right) \) be an \( \\epsilon \) -irregular pair in a graph \( G \), with \( \\mid d\\left( {{X}_{1},{Y}_{1}}\\right) - \) \( d\\left( {X, Y}\\right) \\mid > \\epsilon \), where \( {X}_{1} \) is a large subset of \( X \), and \( {Y}_{1} \) is a large subset of \( Y \) . Define \( {X}_{... | Proof Set\n\n\[ x \\mathrel{\\text{:=}} \\left| X\\right| ,\\;y \\mathrel{\\text{:=}} \\left| Y\\right| ,\\;d \\mathrel{\\text{:=}} d\\left( {X, Y}\\right) \]\n\nand\n\n\[ {x}_{i} \\mathrel{\\text{:=}} \\left| {X}_{i}\\right| ,\\;{y}_{j} = \\left| {Y}_{j}\\right| ,\\;{d}_{ij} \\mathrel{\\text{:=}} d\\left( {{X}_{i},{Y}... | Yes |
Lemma 13.1 The Crossing Lemma\n\nLet \( G \) be a simple graph with \( m \geq {4n} \) . Then\n\n\[ \operatorname{cr}\left( G\right) \geq \frac{1}{64}\frac{{m}^{3}}{{n}^{2}} \] | Proof Consider a planar embedding \( \widetilde{G} \) of \( G \) with \( \operatorname{cr}\left( G\right) \) crossings. Let \( S \) be a random subset of \( V \) obtained by choosing each vertex of \( G \) independently with probability \( p \mathrel{\text{:=}} {4n}/m \), and set \( H \mathrel{\text{:=}} G\left\lbrack ... | Yes |
Theorem 13.2 Let \( P \) be a set of \( n \) points in the plane, and let \( \ell \) be the number of lines in the plane passing through at least \( k{+1} \) of these points, where \( 1 \leq k \leq 2\sqrt{2n} \) . Then \( \ell < {32}{n}^{2}/{k}^{3} \) . | Proof Form a graph \( G \) with vertex set \( P \) whose edges are the segments between consecutive points on the lines which pass through at least \( k + 1 \) points of \( P \) . This graph has at least \( k\ell \) edges and crossing number at most \( \left( \begin{array}{l} \ell \\ 2 \end{array}\right) \) . Thus eith... | Yes |
Theorem 13.3 Let \( P \) be a set of \( n \) points in the plane, and let \( k \) be the number of pairs of points of \( P \) at unit distance. Then \( k < 5{n}^{4/3} \) . | Proof Draw a unit circle around each point of \( P \) . Let \( {n}_{i} \) be the number of these circles passing through exactly \( i \) points of \( P \) . Then \( \mathop{\sum }\limits_{{i = 0}}^{{n - 1}}{n}_{i} = n \) and \( k = \) \( \frac{1}{2}\mathop{\sum }\limits_{{i = 0}}^{{n - 1}}i{n}_{i} \) . Now form a graph... | Yes |
Proposition 13.4 MARKOV'S INEQUALITY\n\nLet \( X \) be a nonnegative random variable and \( t \) a positive real number. Then\n\n\[ P\left( {X \geq t}\right) \leq \frac{E\left( X\right) }{t} \] | Proof\n\n\[ E\left( X\right) = \sum \{ X\left( \omega \right) P\left( \omega \right) : \omega \in \Omega \} \geq \sum \{ X\left( \omega \right) P\left( \omega \right) : \omega \in \Omega, X\left( \omega \right) \geq t\} \]\n\n\[ \geq \sum \{ {tP}\left( \omega \right) : \omega \in \Omega, X\left( \omega \right) \geq t\}... | Yes |
Corollary 13.5 Let \( {X}_{n} \) be a nonnegative integer-valued random variable in a probability space \( \left( {{\Omega }_{n},{P}_{n}}\right), n \geq 1 \) . If \( E\left( {X}_{n}\right) \rightarrow 0 \) as \( n \rightarrow infty \), then \( P\left( {{X}_{n} = 0}\right) \rightarrow 1 \) as \( n \rightarrow \infty \) ... | As a simple example, let \( X \) be the number of triangles in \( G \in {\mathcal{G}}_{n, p} \) . We may express \( X \) as the sum \( X = \sum \left\{ {{X}_{S} : S \subseteq V,\left| S\right| = 3}\right\} \), where \( {X}_{S} \) is the indicator random variable for the event \( {A}_{S} \) that \( G\left\lbrack S\right... | Yes |
Theorem 13.6 A random graph in \( {\mathcal{G}}_{n, p} \) almost surely has stability number at most \( \left\lceil {2{p}^{-1}\log n}\right\rceil \) . | Proof Let \( G \in {\mathcal{G}}_{n, p} \) and let \( S \) be a given set of \( k + 1 \) vertices of \( G \), where \( k \in \mathbb{N} \) . The probability that \( S \) is a stable set of \( G \) is \( {\left( 1 - p\right) }^{\left( \begin{matrix} k + 1 \\ 2 \end{matrix}\right) } \), this being the probability that no... | Yes |
Theorem 13.7 Chebyshev's Inequality\n\nLet \( X \) be a random variable and let \( t \) be a positive real number. Then\n\n\[ P\left( {\left| {X - E\left( X\right) }\right| \geq t}\right) \leq \frac{V\left( X\right) }{{t}^{2}} \] | Proof By Markov's Inequality,\n\n\[ P\left( {\left| {X - E\left( X\right) }\right| \geq t}\right) = P\left( {{\left( X - E\left( X\right) \right) }^{2} \geq {t}^{2}}\right) \leq \frac{E\left( {\left( X - E\left( X\right) \right) }^{2}\right) }{{t}^{2}} = \frac{V\left( X\right) }{{t}^{2}} \] | Yes |
Corollary 13.8 Let \( {X}_{n} \) be a random variable in a probability space \( \left( {{\Omega }_{n},{P}_{n}}\right), n \geq \) 1. If \( E\left( {X}_{n}\right) \neq 0 \) and \( V\left( {X}_{n}\right) \ll {E}^{2}\left( {X}_{n}\right) \), then | Proof Set \( X \mathrel{\text{:=}} {X}_{n} \) and \( t \mathrel{\text{:=}} \left| {E\left( {X}_{n}\right) }\right| \) in Chebyshev’s Inequality, and observe that \( P\left( {{X}_{n} = 0}\right) \leq P\left( {\left| {{X}_{n} - E\left( {X}_{n}\right) }\right| \geq \left| {E\left( {X}_{n}\right) }\right| }\right) \) becau... | Yes |
Theorem 13.12 The Local Lemma\n\nLet \( {A}_{i}, i \in N \), be events in a probability space \( \left( {\Omega, P}\right) \), and let \( {N}_{i}, i \in N \), be subsets of \( N \) . Suppose that, for all \( i \in N \), \n\n i) \( {A}_{i} \) is independent of the set of events \( \left\{ {{A}_{j} : j \notin {N}_{i}}\ri... | Proof We prove (13.15) by induction with respect to the lexicographic order of the pair \( \left( {\left| {R \cup S}\right| ,\left| S\right| }\right) \). \n\n If \( S = \varnothing \), then \( {B}_{S} = \Omega \) and \( \mathop{\prod }\limits_{{i \in S}}\left( {1 - {p}_{i}}\right) = 1 \), so \n\n \[ \nP\left( {{B}_{R} ... | Yes |
Theorem 13.14 The Local Lemma - Symmetric Version\n\nLet \( {A}_{i}, i \in N \), be events in a probability space \( \left( {\Omega, P}\right) \) having a dependency graph with maximum degree \( d \) . Suppose that \( P\left( {A}_{i}\right) \leq 1/\left( {e\left( {d + 1}\right) }\right), i \in N \) . Then \( P\left( {{... | Proof Set \( {p}_{i} \mathrel{\text{:=}} p, i \in N \), in the Local Lemma. Now set \( p \mathrel{\text{:=}} 1/\left( {d + 1}\right) \) in order to maximize \( p{\left( 1 - p\right) }^{d} \) and apply the inequality \( {\left( d/\left( d + 1\right) \right) }^{d} = {\left( 1 - 1/\left( d + 1\right) \right) }^{d} > {e}^{... | Yes |
Theorem 13.15 Let \( H \mathrel{\text{:=}} \left( {V,\mathcal{F}}\right) \) be a hypergraph in which each edge has at least \( k \) elements and meets at most \( d \) other edges. If \( e\left( {d + 1}\right) \leq {2}^{k - 1} \), then \( H \) is 2 - colourable. | Proof Consider a random 2-colouring of \( V \) . For each edge \( F \), denote by \( {A}_{F} \) the event that \( F \) is monochromatic. Then \( P\left( {A}_{F}\right) = 2 \cdot {2}^{-k} = {2}^{1 - k} \) . The result now follows from Theorem 13.14. | Yes |
Corollary 13.16 Let \( H \mathrel{\text{:=}} \left( {V,\mathcal{F}}\right) \) be a \( k \) -uniform \( k \) -regular hypergraph, where \( k \geq 9 \) . Then \( H \) is 2-colourable. | Proof Set \( d \mathrel{\text{:=}} k\left( {k - 1}\right) \) in Theorem 13.15. | Yes |
Theorem 13.17 Let \( D \) be a strict \( k \) -diregular digraph, where \( k \geq 8 \) . Then \( D \) contains a directed even cycle. | Proof Consider a random 2-colouring \( c \) of \( V \) . For each vertex \( v \) of \( D \), denote by \( {A}_{v} \) the event that \( c\left( u\right) = c\left( v\right) \) for all \( u \in {N}^{ + }\left( v\right) \) . For each colour \( i \), we have \( P\left( {A}_{v}\right) = P\left( {{A}_{v} \mid c\left( v\right)... | Yes |
Lemma 13.18 Let \( G = \left( {V, E}\right) \) be a simple graph, and let \( \left\{ {{V}_{1},{V}_{2},\ldots ,{V}_{k}}\right\} \) be a partition of \( V \) into \( k \) sets, each of cardinality at least \( {2e\Delta } \) . Then there is a stable set \( S \) in \( G \) such that \( \left| {S \cap {V}_{i}}\right| = 1,1 ... | Proof By deleting vertices from \( G \) if necessary, we may assume that \( \left| {V}_{i}\right| = t \mathrel{\text{:=}} \) \( \lceil {2e\Delta }\rceil ,1 \leq i \leq k \) . We select one vertex \( {v}_{i} \) at random from each set \( {V}_{i},1 \leq i \leq k \) , and set \( S \mathrel{\text{:=}} \left\{ {{v}_{1},{v}_... | Yes |
Theorem 13.19 Let \( G = \left( {V, E}\right) \) be a simple \( {2r} \) -regular graph with girth at least \( {2e}\left( {{4r} - 2}\right) \) . Then \( \operatorname{la}\left( G\right) = r + 1 \) . | Proof By (13.17), we must show that \( \operatorname{la}\left( G\right) \leq r + 1 \) . We make use of the fact that every regular graph of even degree admits a decomposition into 2-factors (see Exercise 16.4.16).\n\nConsider such a decomposition \( \left\{ {{F}_{1},{F}_{2},\ldots ,{F}_{r}}\right\} \) of \( G \), and l... | Yes |
If \( G \) is a connected graph, and is neither an odd cycle nor a complete graph, then \( \chi \leq \Delta \) . | Proof Suppose first that \( G \) is not regular. Let \( x \) be a vertex of degree \( \delta \) and let \( T \) be a search tree of \( G \) rooted at \( x \) . We colour the vertices with the colours \( 1,2,\ldots ,\Delta \) according to the greedy heuristic, selecting at each step a leaf of the subtree of \( T \) indu... | Yes |
Every digraph \( D \) contains a directed path with \( \chi \) vertices. | Let \( k \) be the number of vertices in a longest directed path of \( D \) . Consider a maximal acyclic subdigraph \( {D}^{\prime } \) of \( D \) . Because \( {D}^{\prime } \) is a subdigraph of \( D \), each directed path in \( {D}^{\prime } \) has at most \( k \) vertices. We \( k \) -colour \( D \) by assigning to ... | Yes |
Theorem 14.6 If \( G \) is \( k \) -critical, then \( \delta \geq k - 1 \) . | Proof By contradiction. Let \( G \) be a \( k \) -critical graph with \( \delta < k - 1 \), and let \( v \) be a vertex of degree \( \delta \) in \( G \) . Because \( G \) is \( k \) -critical, \( G - v \) is \( \left( {k - 1}\right) \) -colourable. Let \( \left\{ {{V}_{1},{V}_{2},\ldots ,{V}_{k - 1}}\right\} \) be a \... | Yes |
Theorem 14.7 No critical graph has a clique cut. | Proof By contradiction. Let \( G \) be a \( k \) -critical graph. Suppose that \( G \) has a clique cut \( S \) . Denote the \( S \) -components of \( G \) by \( {G}_{1},{G}_{2},\ldots ,{G}_{t} \) . Because \( G \) is \( k \) -critical, each \( {G}_{i} \) is \( \left( {k - 1}\right) \) -colourable. Furthermore, because... | Yes |
Theorem 14.9 Let \( G \) be a \( k \) -critical graph with a 2-vertex cut \( \{ u, v\} \), and let \( e \) be a new edge joining \( u \) and \( v \) . Then:\n\n1. \( G = {G}_{1} \cup {G}_{2} \), where \( {G}_{i} \) is a \( \{ u, v\} \) -component of \( G \) of type \( i, i = 1,2 \) ,\n\n2. both \( {H}_{1} \mathrel{\tex... | ## Proof\n\n1. Because \( G \) is critical, each \( \{ u, v\} \) -component of \( G \) is \( \left( {k - 1}\right) \) -colourable. Now there cannot exist \( \left( {k - 1}\right) \) -colourings of these \( \{ u, v\} \) -components all of which agree on \( \{ u, v\} \), as such colourings would together yield a \( \left... | Yes |
Theorem 14.10 For each positive integer \( k \), there exists a graph with girth at least \( k \) and chromatic number at least \( k \) . | Proof Consider \( G \in {\mathcal{G}}_{n, p} \), and set \( t \mathrel{\text{:=}} \left\lceil {2{p}^{-1}\log n}\right\rceil \) . By Theorem 13.6, almost surely \( \alpha \left( G\right) \leq t \) . Let \( X \) be the number of cycles of \( G \) of length less than \( k \) . By linearity of expectation (13.4), \[ E\left... | Yes |
Theorem 14.11 For any positive integer \( k \), there exists a triangle-free \( k \) -chromatic graph. | Proof For \( k = 1 \) and \( k = 2 \), the graphs \( {K}_{1} \) and \( {K}_{2} \) have the required property. We proceed by induction on \( k \) . Suppose that we have already constructed a triangle-free graph \( {G}_{k} \) with chromatic number \( k \geq 2 \) . Let the vertices of \( {G}_{k} \) be \( {v}_{1},{v}_{2},\... | Yes |
Theorem 14.13 A graph \( G \) is perfect if and only if every induced subgraph \( H \) of \( G \) satisfies the inequality\n\n\[ v\left( H\right) \leq \alpha \left( H\right) \omega \left( H\right) \] | Observe that the above inequality is invariant under complementation, because \( v\left( \bar{H}\right) = v\left( H\right) ,\alpha \left( \bar{H}\right) = \omega \left( H\right) \), and \( \omega \left( \bar{H}\right) = \alpha \left( H\right) \) . Theorem 14.13 thus implies the Perfect Graph Theorem (14.12).\n\nThe pro... | No |
Proposition 14.14 Let \( S \) be a stable set in a minimally imperfect graph \( G \) . Then \( \omega \left( {G - S}\right) = \omega \left( G\right) . | Proof We have the following string of inequalities (Exercise 14.4.5).\n\n\[ \omega \left( {G - S}\right) \leq \omega \left( G\right) \leq \chi \left( G\right) - 1 \leq \chi \left( {G - S}\right) = \omega \left( {G - S}\right) \]\n\nBecause the left and right members are the same, equality holds throughout. In particula... | No |
Lemma 14.15 Let \( G \) be a minimally imperfect graph with stability number \( \alpha \) and clique number \( \omega \) . Then \( G \) contains \( {\alpha \omega } + 1 \) stable sets \( {S}_{0},{S}_{1},\ldots ,{S}_{\alpha \omega } \) and \( {\alpha \omega } + 1 \) cliques \( {C}_{0},{C}_{1},\ldots ,{C}_{\alpha \omega ... | Proof Let \( {S}_{0} \) be a stable set of \( \alpha \) vertices of \( G \), and let \( v \in {S}_{0} \) . The graph \( G - v \) is perfect because \( G \) is minimally imperfect. Thus \( \chi \left( {G - v}\right) = \omega \left( {G - v}\right) \leq \omega \left( G\right) \) . This means that for any \( v \in {S}_{0} ... | Yes |
Theorem 14.19 Let \( G \) be a graph, and let \( D \) be an orientation of \( G \) each of whose induced subdigraphs has a kernel. For \( v \in V \), let \( L\left( v\right) \) be an arbitrary list of at least \( {d}_{D}^{ + }\left( v\right) + 1 \) colours. Then \( G \) admits an L-colouring. | Proof By induction on \( n \), the statement being trivial for \( n = 1 \) . Let \( {V}_{1} \) be the set of vertices of \( D \) whose lists include colour 1. (We may assume that \( {V}_{1} \neq \varnothing \) by renaming colours if necessary.) By assumption, \( D\left\lbrack {V}_{1}\right\rbrack \) has a kernel \( {S}... | Yes |
Proposition 14.21 Let \( f \) be a nonzero polynomial over a field \( F \) in the variables \( \mathbf{x} = \left( {{x}_{1},{x}_{2},\ldots ,{x}_{n}}\right) \), of degree \( {d}_{i} \) in \( {x}_{i},1 \leq i \leq n \) . Let \( {L}_{i} \) be a set of \( {d}_{i} + 1 \) elements of \( F,1 \leq i \leq n \) . Then there exis... | Proof As noted above, the case \( n = 1 \) simply expresses the fact that a polynomial of degree \( d \) in one variable has at most \( d \) distinct roots. We proceed by induction on \( n \), where \( n \geq 2 \) . We first express \( f \) as a polynomial in \( {x}_{n} \) whose coefficients \( {f}_{j} \) are polynomia... | Yes |
Theorem 14.22 The Combinatorial Nullstellensatz\n\nLet \( f \) be a polynomial over a field \( F \) in the variables \( \mathbf{x} = \left( {{x}_{1},{x}_{2},\ldots ,{x}_{n}}\right) \) . Suppose that the total degree of \( f \) is \( \mathop{\sum }\limits_{{1 = 1}}^{n}{d}_{i} \) and that the coefficient in \( f \) of \(... | Proof For \( 1 \leq i \leq n \), set\n\n\[ \n{f}_{i} \mathrel{\text{:=}} \mathop{\prod }\limits_{{t \in {L}_{i}}}\left( {{x}_{i} - t}\right) \n\]\n\nThen \( {f}_{i} \) is a polynomial of degree \( \left| {L}_{i}\right| = {d}_{i} + 1 \), with leading term \( {x}_{i}^{{d}_{i} + 1} \), so we may write \( {f}_{i} = {g}_{i}... | Yes |
Corollary 14.23 Let \( G \) be a graph, and let \( D \) be an orientation of \( G \) without directed odd cycles. Then \( G \) is \( \left( {\mathbf{d} + \mathbf{1}}\right) \) -list-colourable, where \( \mathbf{d} \) is the outdegree sequence of \( D \) . | Proof Every orientation of \( G \) with outdegree sequence \( \mathbf{d} \) has the same sign as \( D \) (Exercise 14.6.2a). Therefore \( w\left( \mathbf{d}\right) \neq 0 \) . The result follows on applying Theorem 14.22 with \( f\left( \mathbf{x}\right) = A\left( {G,\mathbf{x}}\right) \) . | No |
Corollary 14.24 If \( G \) has an odd number of orientations \( D \) with outdegree sequence \( \mathbf{d} \), then \( G \) is \( \left( {\mathbf{d} + \mathbf{1}}\right) \) -list-colourable. | Proof In this case \( w\left( \mathbf{d}\right) \) is also odd, thus nonzero. | No |
Theorem 15.1 HEAWOOD'S INEQUALITY\n\nFor any surface \( \sum \) with Euler characteristic \( c \leq 0 \) :\n\n\[ \chi \left( \sum \right) \leq \frac{1}{2}\left( {7 + \sqrt{{49} - {24c}}}\right) \] | As Heawood observed, in order to show that the bound in Theorem 15.1 is attainable for a given surface \( \sum \), it suffices to find just one graph which is embeddable on the surface and requires the appropriate number of colours. Because the inequality \( \chi \leq n \) was employed in deriving Heawood’s Inequality,... | No |
Proposition 15.2 Let \( G \) be a smallest counterexample to the Four-Colour Theorem. Then:\ni) \( G \) is \( 5 \) -critical,\nii) \( G \) is a triangulation,\niii) \( G \) has no vertex of degree less than four. | ## Proof\ni) Clearly, \( G \) must be 5-critical, otherwise there would exist a proper subgraph of \( G \) that is not 4-colourable, contradicting the minimality of \( v\left( G\right) + e\left( G\right) \) .\nii) To see that \( G \) is a triangulation, suppose that it has a face whose boundary is a cycle \( C \) of le... | Yes |
Theorem 15.3 \( G \) has no vertex of degree four. | Proof By contradiction. Let \( v \) be a vertex of degree four in \( G \), and let \( \left\{ {{V}_{1},{V}_{2},{V}_{3}}\right. \) , \( \left. {V}_{4}\right\} \) be a 4-colouring of \( G - v \) ; such a colouring exists because \( G \) is 5-critical. Because \( G \) itself is not 4-colourable, \( v \) must be adjacent t... | Yes |
Theorem 15.8 Let \( G \) be a near-triangulation whose outer face is bounded by a cycle \( C \), and let \( x \) and \( y \) be consecutive vertices of \( C \). Suppose that \( L : V \rightarrow {2}^{\mathbb{N}} \) is an assignment of lists of colours to the vertices of \( G \) such that:\n\ni) \( \left| {L\left( x\rig... | Proof By induction on \( v\left( G\right) \). If \( v\left( G\right) = 3 \), then \( G = C \) and the statement is trivial. So we may assume that \( v\left( G\right) > 3 \).\n\nLet \( z \) and \( {x}^{\prime } \) be the immediate predecessors of \( x \) on \( C \). Consider first the case where \( {x}^{\prime } \) has ... | Yes |
Every 4-chromatic graph contains a \( {K}_{4} \) -subdivision. | Proof Let \( G \) be a 4-chromatic graph, and let \( F \) be a 4-critical subgraph of \( G \) . By Theorem 14.6, \( \delta \left( F\right) \geq 3 \) . By Exercise 10.1.5, \( F \) contains a subdivision of \( {K}_{4} \) , so \( G \) does too. | No |
Theorem 15.13 Every simple graph \( G \) with \( m \geq {2}^{k - 3}n \) has a \( {K}_{k} \) -minor. | Proof By induction on \( m \) . The validity of the theorem is readily verified when \( k \leq 3 \) . Thus we may assume that \( k \geq 4 \) and \( m \geq {10} \) . Let \( G \) be a graph with \( n \) vertices and \( m \) edges, where \( m \geq {2}^{k - 3}n \) . If \( G \) has an edge \( e \) which lies in at most \( {... | Yes |
Corollary 15.14 For \( k \geq 2 \), every \( \left( {{2}^{k - 2} + 1}\right) \) -chromatic graph has a \( {K}_{k} \) -minor. | Proof Let \( G \) be a \( \left( {{2}^{k - 2} + 1}\right) \) -chromatic graph, and let \( F \) be a \( \left( {{2}^{k - 2} + 1}\right) \) -critical subgraph of \( G \) . By Theorem 14.6, \( \delta \left( F\right) \geq {2}^{k - 2} \), and so \( e\left( F\right) \geq {2}^{k - 3}v\left( F\right) \) . Theorem 15.13 implies... | Yes |
Theorem 15.15 Almost every graph is a counterexample to Hajós’ Conjecture. | Proof Let \( G \in {\mathcal{G}}_{n,1/2} \) . Then almost surely \( \alpha \leq \left\lceil {2{\log }_{2}n}\right\rceil \) (Exercise 13.2.11). Thus almost surely\n\n\[ \chi \geq \frac{n}{\alpha } \geq \frac{n}{2{\log }_{2}n} \]\n\n(15.2)\n\nNow the expected number of subgraphs of \( G \) with \( s \) vertices and \( t ... | No |
Problem 16.2 The Assignment Problem\n\nA certain number of jobs are available to be filled. Given a group of applicants for these jobs, fill as many of them as possible, assigning applicants only to jobs for which they are qualified.\n\nThis situation can be represented by means of a bipartite graph \( G\left\lbrack {X... | As we show in Section 16.5, the Assignment Problem can be solved in polynomial time. Indeed, we present there a polynomial-time algorithm for finding a maximum matching in an arbitrary graph. The notions of alternating and augmenting paths with respect to a given matching, defined below, play an essential role in these... | No |
A matching \( M \) in a graph \( G \) is a maximum matching if and only if \( G \) contains no \( M \) -augmenting path. | Proof Let \( M \) be a matching in \( G \) . Suppose that \( G \) contains an \( M \) -augmenting path \( P \) . Then \( {M}^{\prime } \mathrel{\text{:=}} M\bigtriangleup E\left( P\right) \) is a matching in \( G \), and \( \left| {M}^{\prime }\right| = \left| M\right| + 1 \) (see Figure 16.3). Thus \( M \) is not a ma... | Yes |
Theorem 16.4 HALL'S THEOREM\n\nA bipartite graph \( G \mathrel{\text{:=}} G\left\lbrack {X, Y}\right\rbrack \) has a matching which covers every vertex in \( X \) if and only if\n\n\[ \left| {N\left( S\right) }\right| \geq \left| S\right| \text{ for all }S \subseteq X \] | Proof Let \( G \mathrel{\text{:=}} G\left\lbrack {X, Y}\right\rbrack \) be a bipartite graph which has a matching \( M \) covering every vertex in \( X \) . Consider a subset \( S \) of \( X \) . The vertices in \( S \) are matched under \( M \) with distinct vertices in \( N\left( S\right) \) . Therefore \( \left| {N\... | Yes |
Corollary 16.5 A bipartite graph \( G\left\lbrack {X, Y}\right\rbrack \) has a perfect matching if and only if \( \left| X\right| = \left| Y\right| \) and \( \left| {N\left( S\right) }\right| \geq \left| S\right| \) for all \( S \subseteq X \) . | This criterion is satisfied by all nonempty regular bipartite graphs.\n\nCorollary \( \mathbf{{16.6}} \) Every nonempty regular bipartite graph has a perfect matching.\n\nProof Let \( G\left\lbrack {X, Y}\right\rbrack \) be a \( k \) -regular bipartite graph, where \( k \geq 1 \) . Then \( \left| X\right| = \left| Y\ri... | No |
Lemma 16.10 Let \( G \) be a connected graph no vertex of which is essential. Then \( G \) is hypomatchable. | Proof Since no vertex of \( G \) is essential, \( G \) has no perfect matching. It remains to show that every vertex-deleted subgraph has a perfect matching. If this is not so, then each maximum matching leaves at least two vertices uncovered. Thus it suffices to show that for any maximum matching and any two vertices ... | No |
A graph \( G \) has a perfect matching if and only if \[ o\left( {G - S}\right) \leq \left| S\right| \text{ for all }S \subseteq V \] | Proof As already noted,(16.4) holds if \( G \) has a perfect matching. Conversely, let \( G \) be a graph which has no perfect matching. Consider a maximum matching \( {M}^{ * } \) of \( G \), and denote by \( U \) the set of vertices in \( G \) not covered by \( {M}^{ * } \) . By Theorem 16.11, \( G \) has a barrier, ... | Yes |
Theorem 16.14 Petersen's Theorem\n\nEvery 3-regular graph without cut edges has a perfect matching. | Proof We derive Petersen's Theorem from Tutte's Theorem (16.13).\n\nLet \( G \) be a 3-regular graph without cut edges, and let \( S \) be a subset of \( V \) . Consider the vertex sets \( {S}_{1},{S}_{2},\ldots ,{S}_{k} \), of the odd components of \( G - S \) . Because \( G \) has no cut edges, \( d\left( {S}_{i}\rig... | No |
Problem 16.16 The Minimum-Weight Matching Problem\n\nGIVEN: a weighted complete graph \( G \mathrel{\text{:=}} \left( {G, w}\right) \) of even order,\n\nFIND: a minimum-weight perfect matching in \( G \) . | This latter problem can be seen to include the maximum matching problem: it suffices to embed the input graph \( G \) in a complete graph of even order, and assign weight zero to each edge of \( G \) and weight one to each of the remaining edges. Edmonds (1965b) found a polynomial-time algorithm for solving the Minimum... | No |
Consider the bipartite graph in Figure 16.12a, with the indicated matching \( M \) . Figure 16.12b shows an \( M \) -alternating \( {x}_{1} \) -tree, which is grown until the \( M \) -augmenting \( {x}_{1} \) -path \( P \mathrel{\text{:=}} {x}_{1}{y}_{2}{x}_{2}{y}_{1} \) is found. As before, the red vertices are indica... | However, for the purpose of illustrating the entire algorithm, we continue, deleting \( V\left( {T}_{1}\right) \) from \( G \) and growing an \( M \) -alternating \( {y}_{4} \) -tree in the resulting subgraph (see Figure 16.12e), thereby obtaining the APS-tree \( {T}_{2} \) with \( R\left( {T}_{2}\right) = \left\{ {{y}... | Yes |
Theorem 16.21 The matching \( {M}^{ * } \) returned by Egerváry’s Algorithm is a maximum matching. | Proof Let \( \mathcal{T}, R, B \), and \( U \) be the sets of trees, red vertices, blue vertices, and uncovered vertices returned by Egerváry’s Algorithm. Because each tree \( T \in \mathcal{T} \) contains exactly one uncovered vertex, namely its root \( u\left( T\right) \), we have \( \left| U\right| = \left| \mathcal... | Yes |
Proposition 16.22 Let \( F \) be a flower of \( G \) . Then:\ni) \( F \) is connected and of odd order,\nii) for any vertex \( v \) of \( F \), there is an \( M \) -alternating uv-path in \( G \) of even length (that is, one terminating in an edge of \( M \) ). | Proof The proof is by induction on \( i \), where \( F \) is a flower associated with \( {T}_{i} \) . We leave the details to the reader (Exercise 16.5.8). | No |
Corollary 16.23 Let \( {T}_{k} \) be an APS-tree of \( {G}_{k} \) no two red vertices of which are adjacent in \( {G}_{k} \) . Then the red vertices of \( {T}_{k} \) are adjacent in \( {G}_{k} \) only to blue vertices of \( {T}_{k} \) . Equivalently, the flowers of \( G \) associated with \( {T}_{k} \) are adjacent onl... | Proof It follows from Proposition 16.22(ii) that if \( {G}_{k} \) has an \( M \) -augmenting \( u \) - path, then so has \( G \) . Thus if \( G \) has no \( M \) -augmenting \( u \) -path, no red vertex of \( {T}_{k} \) can be adjacent in \( {G}_{k} \) to any vertex in \( V\left( {G}_{k}\right) \smallsetminus V\left( {... | Yes |
In a school, there are \( m \) teachers \( {x}_{1},{x}_{2},\ldots ,{x}_{m} \), and \( n \) classes \( {y}_{1},{y}_{2},\ldots ,{y}_{n} \) . Given that teacher \( {x}_{i} \) is required to teach class \( {y}_{j} \) for \( {p}_{ij} \) periods, schedule a complete timetable in the minimum number of periods. | To solve this problem, we represent the teaching requirements by a bipartite graph \( H\left\lbrack {X, Y}\right\rbrack \), where \( X = \left\{ {{x}_{1},{x}_{2},\ldots ,{x}_{m}}\right\}, Y = \left\{ {{y}_{1},{y}_{2},\ldots ,{y}_{n}}\right\} \), and vertices \( {x}_{i} \) and \( {y}_{j} \) are joined by \( {p}_{ij} \) ... | No |
Theorem 17.2 If \( G \) is bipartite, then \( {\chi }^{\prime } = \Delta \) . | Proof By induction on \( m \) . Let \( e = {uv} \) be an edge of \( G \) . We assume that \( H = G \smallsetminus e \) has a \( \Delta \) -edge-colouring \( \left\{ {{M}_{1},{M}_{2},\ldots ,{M}_{\Delta }}\right\} \) . If some colour is available for \( e \), that colour can be assigned to \( e \) to yield a \( \Delta \... | Yes |
Theorem 17.5 For any graph \( G,{\chi }^{\prime } \leq \Delta + \mu \) . | This more general theorem can be established by adapting the proof of Theorem 17.4 (Exercise 17.2.6). The graph \( G \) depicted in Figure 17.3 shows that the theorem is best possible for any value of \( \mu \) . Here \( \Delta = {2\mu } \) and, the edges being pairwise adjacent, \( {\chi }^{\prime } = m = {3\mu } = \D... | No |
Theorem 17.9 Let \( G\left\lbrack {X, Y}\right\rbrack \) be a simple bipartite graph, and let \( D \) be an orientation of its line graph \( L\left( G\right) \) in which each \( X \) -clique and each \( Y \) -clique induces a transitive tournament. Then \( D \) has a kernel. | Proof By induction on \( e\left( G\right) \), the case \( e\left( G\right) = 1 \) being trivial. For \( v \in V\left( G\right) \) , denote by \( {T}_{v} \) the transitive tournament in \( D \) corresponding to \( v \), and for \( x \in X \) , denote by \( {t}_{x} \) the sink of \( {T}_{x} \) . Set \( K \mathrel{\text{:... | Yes |
Theorem 18.1 Let \( S \) be a set of vertices of a hamiltonian graph \( G \) . Then\n\n\[ c\left( {G - S}\right) \leq \left| S\right| \]\n\n(18.1)\n\nMoreover, if equality holds in (18.1), then each of the \( \left| S\right| \) components of \( G - S \) is traceable, and every Hamilton cycle of \( G \) includes a Hamil... | Proof Let \( C \) be a Hamilton cycle of \( G \) . Then \( C - S \) clearly has at most \( \left| S\right| \) components. But this implies that \( G - S \) also has at most \( \left| S\right| \) components, because \( C \) is a spanning subgraph of \( G \) .\n\nIf \( G - S \) has exactly \( \left| S\right| \) component... | Yes |
Theorem 18.4 Dirac's Theorem\n\nLet \( G \) be a simple graph of minimum degree \( \delta \), where \( \delta \geq n/2 \) and \( n \geq 3 \) . Then \( G \) is hamiltonian. | Proof Form a 2-edge-coloured complete graph \( K \) with vertex set \( V \) by colouring the edges of \( G \) blue and the edges of its complement \( \bar{G} \) red. Let \( C \) be a Hamilton cycle of \( K \) with as many blue edges as possible. We show that every edge of \( C \) is blue, in other words, that \( C \) i... | Yes |
Lemma 18.5 Let \( G \) be a simple graph and let \( u \) and \( v \) be nonadjacent vertices in \( G \) whose degree sum is at least \( n \) . Then \( G \) is hamiltonian if and only if \( G + {uv} \) is hamiltonian. | Proof If \( G \) is hamiltonian, so too is \( G + {uv} \) . Conversely, suppose that \( G + {uv} \) has a Hamilton cycle \( C \) . Then, as in the proof of Theorem 18.4 (with \( x \mathrel{\text{:=}} u \) and \( \left. {{x}^{ + } \mathrel{\text{:=}} v}\right) \), there is a cycle exchange transforming \( C \) to a Hami... | Yes |
Theorem 18.7 A simple graph is hamiltonian if and only if its closure is hamiltonian. | Proof Apply Lemma 18.5 each time an edge is added in the formation of the closure. | No |
Theorem 18.9 Let \( G \) be a simple graph with degree sequence \( \left( {{d}_{1},{d}_{2},\ldots ,{d}_{n}}\right) \) , where \( {d}_{1} \leq {d}_{2} \leq \cdots \leq {d}_{n} \) and \( n \geq 3 \) . Suppose that there is no integer \( k < n/2 \) such that \( {d}_{k} \leq k \) and \( {d}_{n - k} < n - k \) . Then \( G \... | Proof Let \( {G}^{\prime } \) be the closure of \( G \) . We show that \( {G}^{\prime } \) is complete. The conclusion then follows from Corollary 18.8. We denote the degree of a vertex \( v \) in \( {G}^{\prime } \) by \( {d}^{\prime }\left( v\right) \) .\n\nAssume, to the contrary, that \( {G}^{\prime } \) is not com... | Yes |
Theorem 18.10 The Chvátal-Erdős Theorem\n\nLet \( G \) be a graph on at least three vertices with stability number \( \alpha \) and connectivity \( \kappa \), where \( \alpha \leq \kappa \). Then \( G \) is hamiltonian. | Proof Let \( C \) be a longest cycle in \( G \). Suppose that \( C \) is not a Hamilton cycle. Let \( B \) be a proper bridge of \( C \) in \( G \), and denote by \( S \) its set of vertices of attachment to \( C \). For any two vertices \( x \) and \( y \) of \( S \), there is a path \( {xPy} \) in \( B \); this path ... | Yes |
Theorem 18.11 The Lollipop Lemma\n\nLet \( G \) be a connected graph on at least two vertices, and let \( x \) be a vertex of \( G \) . Then the number of longest \( x \) -paths of \( G \) that terminate in a vertex of even degree is even. | Proof Let \( H \) denote the \( x \) -path graph of \( G \) and let \( P \) be a longest \( x \) -path of \( G \) . If \( P \) terminates in \( y \) ,\n\n\n\nFig. 18.19. (a) A graph \( G \) ,(b) the \( x \) -path gra... | Yes |
Corollary 18.12 Let \( G \) be a graph on at least three vertices, and let \( x \) and \( y \) be two vertices of \( G \) . Suppose that each vertex of \( G \) other than \( x \) and \( y \) is of odd degree. Then the number of Hamilton xy-paths in \( G \) is even. In particular, if \( G \) is a graph in which all vert... | Proof We may assume that \( G \) has at least one Hamilton \( {xy} \) -path, otherwise the conclusion is trivial. Set \( {G}^{\prime } \mathrel{\text{:=}} G - y \) . The longest \( x \) -paths of \( {G}^{\prime } \) are Hamilton paths of \( {G}^{\prime } \), and each Hamilton \( {xy} \) -path of \( G \) is an extension... | Yes |
Theorem 18.15 Let \( G \) be a graph and let \( C \) be a Hamilton cycle of \( G \) . Colour the edges of \( C \) red and the remaining edges of \( G \) blue. Suppose that there is a red-stable blue-dominating set \( S \) in \( G \) . Then \( G \) has a second Hamilton cycle. | Proof Let \( S \) be a red-stable blue-dominating set, and let \( T \mathrel{\text{:=}} {S}^{ - } \cup {S}^{ + } \) . Consider the spanning subgraph \( H \) of \( G \) the edge set of which consists of all red edges and, for each vertex \( y \in T \), one blue edge joining \( y \) to a vertex of \( S \) . In \( H \), e... | Yes |
Corollary 18.16 Let \( G\left\lbrack {X, Y}\right\rbrack \) be a simple hamiltonian bipartite graph in which each vertex of \( Y \) has degree three or more. Then \( G \) has at least two Hamilton cycles. | Proof Let \( C \) be a Hamilton cycle of \( G \) . Colour the edges of \( C \) red and the remaining edges of \( G \) blue. Each vertex of \( Y \) is then incident to at least one blue edge, so \( X \) is a red-stable blue-dominating set in \( G \) . By Theorem 18.15, \( G \) has a second Hamilton cycle. | Yes |
Theorem 18.17 For \( k \geq {73} \), every simple hamiltonian \( k \) -regular graph has at least two Hamilton cycles. | Proof Let \( G \) be a simple hamiltonian \( k \) -regular graph, and let \( C \) be a Hamilton cycle of \( G \) . As in Theorem 18.15, we colour the edges of \( C \) red and the remaining edges of \( G \) blue. We now select each vertex of \( G \) independently, each with probability \( p \), so as to obtain a random ... | Yes |
Theorem 18.19 PósA's LEMMA\n\nLet \( {xPy} \) be a longest path in a graph \( G \) . Denote by \( \mathcal{P} \) the set of all \( x \) -paths of \( G \) obtainable from \( P \) by path exchanges, by \( T \) the set of all terminal vertices of paths in \( \mathcal{P} \), and by \( {T}^{ - } \) and \( {T}^{ + } \), resp... | Proof Let \( u \in S \) and \( v \in T \) . By the definition of \( T \), there exists a path \( {xQv} \) in \( \mathcal{P} \) . If \( u \in V\left( G\right) \smallsetminus V\left( P\right) \), then \( u \) and \( v \) cannot be adjacent because \( Q \) is a longest path in \( G \) . So suppose that \( u \in V\left( P\... | Yes |
Corollary 18.20 Let \( G \in {\mathcal{G}}_{n, p} \), where \( p = 9\log n/n \), and let \( T \) be as defined in the statement of Pósa’s Lemma. Then \( P\left( {\left| T\right| < \lfloor n/4\rfloor }\right) \ll {n}^{-1} \) . | Proof Suppose that \( \left| T\right| = k \) . Then \( \left| {T}^{ - }\right| \leq k \) and \( \left| {T}^{ + }\right| \leq k \) . By Theorem 18.19, there is therefore a subset \( S \) of \( V \), disjoint from \( T \), such that \( \left| S\right| \geq n - 1 - {3k} \) and \( e\left( {S, T}\right) = 0 \) . The probabi... | Yes |
Theorem 18.21 Let \( G \in {\mathcal{G}}_{n, p} \), where \( p = 9\log n/n \) . Then \( G \) is almost surely traceable. | Proof For \( v \in V \), the vertex-deleted subgraph \( G - v \) is a random graph on \( n - 1 \) vertices with independent edge probability \( p \) . Let \( {T}_{v} \) be the set \( T \) as defined in the statement of Pósa’s Lemma, applied to this random graph \( G - v \) . We consider the following two events.\n\n\( ... | Yes |
Theorem 18.22 Let \( G \in {\mathcal{G}}_{n, p} \), where \( p = {10}\log n/n \) . Then \( G \) is almost surely hamiltonian. | Proof Let \( H = {G}_{1} \cup {G}_{2} \), where \( {G}_{i} \in {\mathcal{G}}_{n,{p}_{i}}, i = 1,2 \), with \( {p}_{1} = 9\log n/n \) and \( {p}_{2} = \log n/n \) . Then \( H \in {\mathcal{G}}_{n, p} \), where \( p = {10}\left( {\log n/n}\right) - 9{\left( \log n/n\right) }^{2} \) . It suffices to show that \( H \) is a... | Yes |
Theorem 19.1 The Erdős-Pósa Theorem\n\nFor any positive integer \( k \), a graph either contains \( k \) disjoint cycles or else has a set of at most \( {4k}\log k \) vertices whose deletion destroys all cycles. | It was shown by Robertson and Seymour (1986) that, for fixed \( k \), this theorem yields a linear-time algorithm for deciding whether an input graph \( G \) has a family of \( k \) disjoint cycles. For if \( G \) does not contain \( k \) disjoint cycles, there is a set of at most \( {4k}\log k \) vertices whose deleti... | No |
Theorem 19.3 Let \( \mathcal{P} \) be an optimal path partition of a digraph \( D \) . Then there is a stable set \( S \) in \( D \) which is orthogonal to \( \mathcal{P} \) . | Note that the Gallai-Milgram Theorem is an immediate consequence of Theorem 19.3 because \( \pi = \left| \mathcal{P}\right| \leq \left| S\right| \leq \alpha \) . Theorem 19.3 is established by means of an inductive argument involving the sets of initial and terminal vertices of the constituent paths of a path partition... | No |
Theorem 19.5 Dilworth's Theorem\n\nThe minimum number of chains into which the elements of a partially ordered set \( P \) can be partitioned is equal to the maximum number of elements in an antichain of \( P \) . | Proof Let \( P \mathrel{\text{:=}} \left( {X, \prec }\right) \), and denote by \( D \mathrel{\text{:=}} D\left( P\right) \) the digraph whose vertex set is \( X \) and whose arcs are the ordered pairs \( \left( {u, v}\right) \) such that \( u \prec v \) in \( P \) . Chains and antichains in \( P \) correspond in \( D \... | Yes |
Theorem 19.9 Every strong digraph admits a coherent cyclic order. | Proof Let \( D \) be a strong digraph. By Theorem 5.14, \( D \) has a coherent feedback arc set \( S \) . Since \( D \smallsetminus S \) is acyclic, the vertices of \( D \) may be ordered as \( {v}_{1},{v}_{2},\ldots ,{v}_{n} \) , so that every arc not in \( S \) joins some vertex \( {v}_{i} \) to a vertex \( {v}_{j} \... | No |
Proposition 19.16 In any digraph, the size of a maximum 2-packing of directed bonds is equal to twice the size of a maximum packing of directed bonds. | Proof To establish the required equality, it is enough to show that any 2-packing \( \mathcal{B} \) of directed bonds contains a packing consisting of at least half its members.\n\nRecall that subsets \( X \) and \( Y \) of \( V \) cross if \( X \cap Y, X \smallsetminus Y, Y \smallsetminus X \), and \( V \smallsetminus... | Yes |
In any digraph, the maximum number of arc-disjoint directed bonds is equal to the minimum number of arcs which meet all directed bonds. | When \( \nu \left( D\right) = 0 \), the digraph \( D \) has no directed bonds, and equality clearly holds. Suppose that the theorem is false. Then there is a smallest positive integer \( k \), and a digraph \( D \), such that:\n\n\[ \nu \left( D\right) = k\;\text{ and }\;\tau \left( D\right) > k \]\n\n(19.8)\n\nDenote ... | No |
Proposition 20.1 The circulation space \( \mathcal{C} \) of a digraph \( D \) is the orthogonal complement of the row space of its incidence matrix \( \mathbf{M} \) . | We now turn our attention to the row space of \( \mathbf{M} \) . Let \( \mathbf{g} \) be an element of the row space, so that \( \mathbf{g} = \mathbf{{pM}} \) for some vector \( \mathbf{p} \in {\mathbb{R}}^{V} \) . Consider an arc \( a \mathrel{\text{:=}} \left( {x, y}\right) \) . In the column \( a \) of \( \mathbf{M}... | No |
Lemma 20.2 Let \( g \) be a nonzero tension in a digraph \( D \) . Then the support of \( g \) contains a bond. Moreover, if \( g \) is nonnegative, then the support of \( g \) contains a directed bond. | Proof Let \( \mathbf{g} \mathrel{\text{:=}} \mathbf{{pM}} \) be a nonzero tension in a digraph \( D \), with support \( S \), and let \( \left( {x, y}\right) \in S \) . Set \( X \mathrel{\text{:=}} \{ v \in V : p\left( v\right) = p\left( x\right) \} \) . Then \( \left( {x, y}\right) \in \partial \left( X\right) \) and ... | Yes |
Theorem 20.6 Let \( \mathbf{B} \) and \( \mathbf{C} \) be basis matrices of \( \mathcal{B} \) and \( \mathcal{C} \), respectively, and let \( S \subseteq A \) . Then:\ni) the columns of \( {\left. \mathbf{B}\right| }_{S} \) are linearly independent if and only if \( S \) contains no cycle,\nii) the columns of \( \mathb... | Proof Denote the column of \( \mathbf{B} \) corresponding to arc \( a \) by \( \mathbf{b}\left( a\right) \) . The columns of \( \mathbf{B} \mid \mathrm{S} \) are linearly dependent if and only if there exists a function \( f \) on \( A \) such that \( \mathop{\sum }\limits_{{a \in A}}f\left( a\right) \mathbf{b}\left( a... | Yes |
Theorem 20.7 The dimensions of the tension and circulation spaces of a connected digraph \( D \) are given by the formulae:\n\n\[ \dim \mathcal{B} = n - 1 \]\n\n(20.4)\n\n\[ \dim \mathcal{C} = m - n + 1 \]\n\n(20.5) | Proof Consider a basis matrix \( \mathbf{B} \) of \( \mathcal{B} \) . By Theorem 20.6,\n\n\[ \operatorname{rank}\mathbf{B} = \max \{ \left| S\right| : S \subseteq A, S\text{ acyclic }\} \]\n\nThe above maximum is attained when \( S \) is a spanning tree of \( D \), and is therefore equal to \( n - 1 \) . Because \( \di... | Yes |
Theorem 20.9 Hoffman's Circulation Theorem\n\nA digraph \( D \) has a feasible circulation with respect to bounds \( b \) and \( c \) if and only if these bounds satisfy inequality (20.6). Furthermore, if both \( b \) and \( c \) are integer-valued and satisfy this inequality, then \( D \) has an integer-valued feasibl... | Now let \( g \) be a feasible tension in \( D \) with respect to \( b \) and \( c \) . Consider any cycle \( C \) of \( D \), and a sense of traversal of \( C \) . Because \( g \) is a tension, \( g\left( {C}^{ + }\right) = g\left( {C}^{ - }\right) \) (Exercise 20.1.1). Moreover, because \( g \) is feasible, \( c\left(... | No |
A digraph \( D \) has a feasible tension with respect to bounds \( b \) and \( c \) if and only if these bounds satisfy inequality (20.7). Furthermore, if both \( b \) and \( c \) are integer-valued and satisfy this inequality, then \( D \) has an integer-valued feasible tension. | Hoffman's Circulation Theorem and Ghouila-Houri's Theorem may both be proved with the aid of a fundamental tool in linear algebra known as Farkas' Lemma (see inset).\n\n## Proof Technique: Farkas' Lemma\n\nA system \( \mathbf{{Ax}} = \mathbf{0} \) of linear equations always has at least one solution. However, in many p... | No |
Proposition 20.12 Let \( D \) be a digraph, and let \( \mathbf{b} \) be a real-valued function defined on \( A \) . Then either there is a circulation \( \mathbf{f} \) in \( D \) such that \( \mathbf{f} \geq \mathbf{b} \), or there is a nonnegative tension \( \mathbf{g} \) in \( D \) such that \( \mathbf{g}\mathbf{b} >... | Proof Consider the incidence matrix \( \mathbf{M} \) of \( D \), and the two linear systems:\n\n\[ \mathbf{{Mf}} = \mathbf{0},\;\mathbf{f} \geq \mathbf{b} \]\n\n(20.10)\n\n\[ \mathbf{{pM}} \geq \mathbf{0},\;\mathbf{{pMb}} > 0 \]\n\n(20.11)\n\nBy Farkas' Lemma, exactly one of these two systems has a solution. The propos... | Yes |
Theorem 20.14 Let \( D \) be a connected digraph and \( \mathbf{B} \) a unimodular basis matrix of its tension space \( \mathcal{B} \). Then\n\n\[ t\left( G\right) = \det {\mathbf{{BB}}}^{t} \] | Proof Using the Cauchy-Binet Formula \( {}^{1} \) for the determinant of the product of two rectangular matrices, we obtain\n\n\[ \det {\mathbf{{BB}}}^{t} = \sum \left\{ {{\left( \det \left( \mathbf{B} \mid \mathrm{S}\right) \right) }^{2} : \mathrm{S} \subseteq \mathrm{A},\left| \mathrm{S}\right| = \mathrm{n} - 1}\righ... | Yes |
Corollary 20.16 Let \( G \) be a loopless connected graph, \( \mathbf{C} \) its conductance matrix, and \( D \) an orientation of \( G \) . Then:\ni) \( \mathbf{C} = {\mathbf{{MM}}}^{t} \), where \( \mathbf{M} \) is the incidence matrix of \( D \) ,\nii) every cofactor of \( \mathbf{C} \) has value \( t\left( G\right) ... | The proof of the following theorem is analogous to that of Theorem 20.14. | No |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.