Q stringlengths 27 2.03k | A stringlengths 4 2.98k | Result stringclasses 2
values |
|---|---|---|
Theorem 17 A graph is planar iff it does not contain a subdivision of \( {K}_{5} \) or \( {K}_{3,3} \) . | Null | No |
Theorem 18 A graph is planar iff it contains neither \( {K}_{5} \) nor \( {K}_{3,3} \) as a minor. | Null | No |
Theorem 19 Let \( R \) be a commutative ring and let the matrices \( {A}_{1},{A}_{2},\ldots ,{A}_{2k} \) be in \( {M}_{k}\left( R\right) \) . Then \( \left\lbrack {{A}_{1},{A}_{2},\ldots ,{A}_{2k}}\right\rbrack = 0 \) . | Proof. We shall deduce the result from a lemma about Euler trails in directed multigraphs. Let \( \overrightarrow{G} \) be a directed multigraph of order \( n \) with edges \( {e}_{1},{e}_{2},\ldots ,{e}_{m} \) . Thus to each edge \( {e}_{i} \) we associate an ordered pair of not necessarily distinct vertices: the init... | Yes |
Lemma 20 If \( m \geq {2n} \) then \( \varepsilon \left( {\overrightarrow{G};x, y}\right) = 0 \) . | Proof of Lemma 20. We may clearly assume that \( \overrightarrow{G} \) has no isolated vertices. Let \( {\overrightarrow{G}}^{\prime } \) be obtained from \( \overrightarrow{G} \) by adding to it a vertex \( {x}^{\prime } \), a path of length \( m + 1 - {2n} \) from \( {x}^{\prime } \) to \( x \), and an edge from \( y... | Yes |
Theorem 2 There is a distribution of currents satisfying Ohm's law and Kirchhoff’s laws in which a current of size 1 enters at \( s \) and leaves at \( t \) . The value of the current in an edge \( {ab} \) is given by \( \left\{ {{N}^{ * }\left( {s, a, b, t}\right) - {N}^{ * }\left( {s, b, a, t}\right) }\right\} /{N}^{... | Null | No |
Corollary 3 If the conductances of the edges are rational and a current of size 1 goes through the network then the current in each edge has rational value. | Null | No |
Theorem 4 If a rectangle can be tiled with squares then the ratio of two neighbouring sides of the rectangle is rational. | Null | No |
Theorem 5 A rectangle has a perfect squaring if, and only if, the ratio of two neighbouring sides is rational. | The result can be proved by putting together appropriate perfect rectangles; for the proof we refer the reader to the original paper of Sprague. | No |
Theorem 6 Let a rectangle \( T \) be tiled with rectangles \( {T}_{1},\ldots ,{T}_{\ell } \) . If each \( {T}_{i} \) has an integer side then so does \( T \) . | First Proof. Construct a bipartite graph \( G \), with vertex classes \( L \) and \( R \), as follows. Let \( L \) (for ’left’ or ’lattice points’) be the set of integer lattice points in the tiled rectangle: \( L = \left\{ {\left( {x, y}\right) \in {\mathbb{Z}}^{2} : 0 \leq x \leq a,0 \leq y \leq b}\right\} \), and le... | Yes |
Theorem 7 Let a box \( B \) in \( {\mathbb{R}}^{n} \) be tiled with boxes \( {B}_{1},\ldots ,{B}_{\ell } \) . If each \( {B}_{i} \) has at least \( k \) integer sides, then \( B \) itself has at least \( k \) integer sides. | Proof. The fourth proof above carries over, mutatis mutandis. Defining \( {\phi }_{\varepsilon } \) as before, with \( {\phi }_{\varepsilon }\left( \mathbf{z}\right) = \left( {{\phi }_{\varepsilon }\left( {z}_{1}\right) ,\ldots ,{\phi }_{\varepsilon }\left( {z}_{n}\right) }\right) \), we find that, if \( \varepsilon > ... | Yes |
Theorem 8 Let \( {a}_{1},\ldots ,{a}_{n} \) be natural numbers with \( {a}_{1}\left| {{a}_{2},\ldots ,{a}_{n - 1}}\right| {a}_{n} \), and let \( B \) be an \( {A}_{1} \times \cdots \times {A}_{n} \) box filled with \( {a}_{1} \times \cdots \times {a}_{n} \) bricks standing in any position. Then B can also be filled wit... | Proof. By Theorem 6, we know that \( {a}_{n} \) divides an \( {A}_{i} \) : let \( \pi \left( n\right) \) be such that \( {a}_{n} \) divides \( {A}_{\pi \left( n\right) } \) . Next, we know by Theorem 7 that \( {a}_{n - 1} \) divides at least two \( {A}_{i} \) : let \( \pi \left( {n - 1}\right) \neq \pi \left( n\right) ... | Yes |
Theorem 10 Let \( D = \left( {D}_{ij}\right) \) be the \( n \times n \) diagonal matrix with \( {D}_{ii} = d\left( {v}_{i}\right) \), the degree of \( {v}_{i} \) in \( G \) . Then \[ B{B}^{t} = D - A\text{. } \] | Proof. What is \( {\left( B{B}^{t}\right) }_{ij} \) ? It is \( \mathop{\sum }\limits_{{l = 1}}^{m}{b}_{il}{b}_{jl} \), which is \( d\left( {v}_{i}\right) \) if \( i = j, - 1 \) if \( {v}_{i}{v}_{j} \) is an edge (if \( {e}_{l} = {v}_{i}{v}_{j} \) is directed from \( {v}_{i} \) to \( {v}_{j} \), then \( {b}_{il}{b}_{jl}... | Yes |
Theorem 11 The electric current \( \mathbf{w} \) satisfying \( \mathbf{p} = R\mathbf{w} + \mathbf{g} \) is given by \( \mathbf{w} = \) \( - C{\left( {C}^{t}RC\right) }^{-1}{C}^{t}\mathbf{g} \) . | Proof. Equation (1) implies that \( {B}_{T}{\mathbf{w}}_{T} + {B}_{N}{\mathbf{w}}_{N} = \mathbf{0} \), so \( {\mathbf{w}}_{T} = - {B}_{T}^{-1}{B}_{N}{\mathbf{w}}_{N} = \) \( {C}_{T}{\mathbf{w}}_{N} \) . Hence \( \mathbf{w} = C{\mathbf{w}}_{N} \) . Combining (2) and (3) we find that \( {C}^{t}R\mathbf{w} + {C}^{t}\mathb... | Yes |
Theorem 12 With the notation above, \( {N}^{ * }\left( G\right) = {K}^{ * }\left( G\right) \) . | Proof. We may assume that \( G \) is connected, since otherwise \( {N}^{ * }\left( G\right) = {K}^{ * }\left( G\right) = \) 0 . Also, the result is trivial for \( n = 1 \) since then \( {N}^{ * }\left( G\right) = {K}^{ * }\left( G\right) = 1 \) .\n\nLet us apply induction on the number of edges of \( G \) . As the resu... | Yes |
Theorem 14 Let \( G \) be a directed multigraph with vertex set \( V\left( G\right) = \left\{ {{v}_{1},\ldots ,{v}_{n}}\right\} \) . For \( 1 \leq i \leq n \), denote by \( {t}_{i}\left( G\right) \) the number of spanning trees oriented towards \( {v}_{i} \) . Also, let \( L = \left( {\ell }_{ij}\right) \) be the combi... | The proof is entirely along the lines of the proof of Theorem 12: when considering \( {t}_{1}\left( G\right) \), say, all we have to take care is to contract all edges from \( {v}_{i} \) to \( {v}_{1} \) for some \( i > 1 \) . Note that this result contains Corollary 13: given a multigraph, replace each edge by two edg... | No |
Theorem 14 Let \( G \) be a directed multigraph with vertex set \( V\left( G\right) = \left\{ {{v}_{1},\ldots ,{v}_{n}}\right\} \) . For \( 1 \leq i \leq n \), denote by \( {t}_{i}\left( G\right) \) the number of spanning trees oriented towards \( {v}_{i} \) . Also, let \( L = \left( {\ell }_{ij}\right) \) be the combi... | The proof is entirely along the lines of the proof of Theorem 12: when considering \( {t}_{1}\left( G\right) \), say, all we have to take care is to contract all edges from \( {v}_{i} \) to \( {v}_{1} \) for some \( i > 1 \) . Note that this result contains Corollary 13: given a multigraph, replace each edge by two edg... | Yes |
Theorem 1 (Max-Flow Min-Cut Theorem.) The maximal flow value from s to t is equal to the minimum of the capacities of cuts separating \( s \) from \( t \) . | Proof. We have remarked already that there is a flow \( f \) with maximal value, say \( v \), and the capacity of every cut is at least \( v \) . Thus, in order to prove the theorem we have to show that there is a cut with capacity \( v \) . We shall, in fact, do considerably more than this: we shall give a very simple... | Yes |
Theorem 4 Let \( \overrightarrow{G} \) be a directed graph with capacity bounds on the vertices other than the source \( s \) and the sink \( t \) . Then the minimum of the capacity of a vertex-cut is equal to the maximum of the flow value from \( s \) to \( t \) . | Null | No |
Theorem 5 (i) Let \( s \) and \( t \) be distinct nonadjacent vertices of a graph \( G \) . Then the minimal number of vertices separating \( s \) from \( t \) is equal to the maximal number of independent \( s - t \) paths. | Proof. (i) Replace each edge \( {xy} \) of \( G \) by two directed edges, \( \overrightarrow{xy} \) and \( \overrightarrow{yx} \), and give each vertex other than \( s \) and \( t \) capacity 1 . Then by Theorem 4 the maximal flow value from \( s \) to \( t \) is equal to the minimum of the capacity of a cut separating... | Yes |
Corollary 6 For \( k \geq 2 \), a graph is \( k \) -connected iff it has at least two vertices and any two vertices can be joined by \( k \) independent paths. Also, for \( k \geq 2 \), a graph is \( k \) -edge-connected iff it has at least two vertices and any two vertices can be joined by \( k \) edge disjoint paths. | Null | No |
Theorem 7 A bipartite graph \( G \) with vertex sets \( {V}_{1} \) and \( {V}_{2} \) contains a complete matching from \( {V}_{1} \) to \( {V}_{2} \) iff\n\n\[ \left| {\Gamma \left( S\right) }\right| \geq \left| S\right| \text{ for every }S \subset {V}_{1}. \] | We have already seen that the condition is necessary so we have to prove only the sufficiency.\n\nFirst Proof. Both Menger’s theorem (applied to the sets \( {V}_{1} \) and \( {V}_{2} \) as at the end of Section 2) and the max-flow min-cut theorem (applied to the directed graph obtained from \( G \) by sending each edge... | Yes |
Theorem 8 A family \( \mathcal{A} = \left\{ {{A}_{1},{A}_{2},\ldots ,{A}_{m}}\right\} \) of sets has a set of distinct representatives iff\n\n\[ \left| {\mathop{\bigcup }\limits_{{i \in F}}{A}_{i}}\right| \geq \left| F\right| \text{ for every }F \subset \{ 1,2,\ldots, m\} \] | Null | No |
Corollary 9 Suppose that a bipartite graph \( G = {G}_{2}\left( {m, n}\right) \), with vertex sets \( {V}_{1},{V}_{2} \), satisfies the following condition:\n\n\[ \left| {\Gamma \left( S\right) }\right| \geq \left| S\right| - d \]\n\nfor every \( {S}_{1} \subset {V}_{1} \) . Then \( G \) contains \( m - d \) independen... | Proof. Add \( d \) vertices to \( {V}_{2} \) and join them to each vertex in \( {V}_{1} \) . The new graph \( G * \) satisfies the conditions for a complete matching. At least \( m - d \) of the edges in a complete matching of \( G * \) belong to \( G \) . | Yes |
Corollary 10 Let \( G = {G}_{2}\left( {m, n}\right) \) be a bipartite graph. Write \( h \) for the maximal number of independent edges, i for the maximal number of independent vertices, and \( j \) for the minimal number of edges and vertices covering all the vertices. Then\n\n\[ i = j = m + n - h. \] | Proof. Let \( {E}^{\prime } \cup {V}^{\prime } \) be a set of \( j \) edges and vertices covering all vertices, with \( {E}^{\prime } \subset E \) and \( {V}^{\prime } \subset V \) . If \( e, f \in {E}^{\prime } \) share a vertex, then in the cover \( {E}^{\prime } \cup {V}^{\prime } \) we may replace \( f \) by its ot... | Yes |
Corollary 11 Let \( G \) be a bipartite graph with vertex classes \( {V}_{1} = \left\{ {{x}_{1},\ldots ,{x}_{m}}\right\} \) and \( {V}_{2} = \left\{ {{y}_{1},\ldots ,{y}_{n}}\right\} \) . Then \( G \) contains a subgraph \( H \) such that \( {d}_{H}\left( {x}_{i}\right) = {d}_{i} \) and \( 0 \leq {d}_{H}\left( {y}_{j}\... | Proof. Replace each vertex \( {x}_{i} \) by \( {d}_{i} \) vertices joined to every vertex in \( \Gamma \left( {x}_{i}\right) \) . Then \( G \) has such a subgraph \( H \) iff the new graph has a matching from the new first vertex class to \( {V}_{2} \) . The result follows from Theorem 7. | Yes |
Theorem 12 Let \( G = {G}_{2}\left( {m, n}\right) \) be a bipartite graph with vertex classes \( {V}_{1} = \) \( \left\{ {{x}_{1},\ldots ,{x}_{m}}\right\} \) and \( {V}_{2} = \left\{ {{y}_{1},\ldots ,{y}_{n}}\right\} \) . For \( S \subset {V}_{1} \) and \( 1 \leq j \leq n \) denote by \( {S}_{j} \) the number of edges ... | Proof. Turn \( G \) into a directed graph \( \overrightarrow{G} \) by sending each edge from \( {V}_{1} \) to \( {V}_{2} \) . Give each edge capacity 1, a vertex \( {x}_{i} \) capacity \( {d}_{i} \), and a vertex \( {y}_{j} \) capacity \( {e}_{j} \) . Then there is a subgraph \( H \) with the required properties iff in... | Yes |
Theorem 13 If every antichain in a (finite) partially ordered set \( P \) has at most \( m \) elements, then \( P \) is the union of \( m \) chains. | Proof. Let us apply induction on \( \left| P\right| \) . If \( P = \varnothing \), there is nothing to prove, so we suppose that \( \left| P\right| > 0 \) and the theorem holds for sets with fewer elements.\n\nLet \( C \) be a maximal chain in \( P \) . (Thus if \( x \notin C \), then \( C \cup \{ x\} \) is no longer a... | Yes |
Corollary 15 A graph \( G \) contains a set of independent edges covering all but at most \( d \) of the vertices iff\n\n\[ q\left( {G - S}\right) \leq \left| S\right| + d \] \n\nfor every \( S \subset V\left( G\right) \) . | Proof. Since the number of vertices not covered by a set of independent edges is congruent to \( \left| G\right| \) modulo 2, we may assume that\n\n\[ d \equiv \left| G\right| \left( {\;\operatorname{mod}\;2}\right) \]\n\nPut \( H = G + {K}_{d} \) ; that is, let \( H \) be obtained from \( G \) by adding to it a set \(... | Yes |
Theorem 16 For every assignment of preferences in a bipartite graph, there is a stable matching. | Proof. Let us describe a variant of the fundamental algorithm we have just mentioned, in which all boys and all girls act simultaneously, in rounds. In every odd round \( \left( {1\mathrm{{st}},3\mathrm{{rd}},\ldots }\right) \), each boy proposes to his highest preference among those girls whom he knows and who have no... | Yes |
Lemma 17 Let \( M \) and \( {M}^{\prime } \) be two stable matchings in a bipartite graph with certain preferences, and let \( C \) be a component of the subgraph \( H \) formed by the edges of \( M \cup {M}^{\prime } \) . If \( C \) has at least three vertices, then it is a preference-oriented cycle. In particular, if... | Proof. In this proof it is best not to distinguish between boys and girls: we shall write \( {x}_{1},{x}_{2},\ldots \) for either of them. We know that \( C \) is either a path of length at least two or a cycle of length at least four.\n\nSuppose that \( C \) contains a path \( {x}_{1}{x}_{2}{x}_{3}{x}_{4} \), with \( ... | Yes |
Theorem 18 For every assignment of preferences in a bipartite graph with bipartition \( \left( {{V}_{1},{V}_{2}}\right) \), there are subsets \( {U}_{1} \subset {V}_{1} \) and \( {U}_{2} \subset {V}_{2} \) such that every stable matching is a complete matching from \( {U}_{1} \) to \( {U}_{2} \) . In particular, all st... | Proof. Suppose that the assertion fails. Then we may assume that some edge \( {aA} \) of \( M \) is such that \( a \) is not incident with any edge of \( {M}^{\prime } \) . As \( {M}^{\prime } \) is a maximal matching, \( {bA} \in {M}^{\prime } \) for some \( b \in {V}_{1}, b \neq a \) . But then the component of \( a ... | No |
Corollary 19 Let \( M \) and \( {M}^{\prime } \) be stable matchings in a bipartite graph, with some assignment of preferences. Suppose \( {aB} \in M \) and \( {aB} \notin {M}^{\prime } \) . Then in \( {M}^{\prime } \) both a and \( B \) have mates; also, one of \( a \) and \( B \) is better off in \( {M}^{\prime } \) ... | Null | No |
Theorem 20 For every assignment of preferences in an \( n \) by \( n \) complete bipartite graph with bipartition \( \left( {{V}_{1},{V}_{2}}\right) \), there is a \( {V}_{1} \) -optimal stable matching. | Proof. Let us denote by \( S\left( a\right) \) the set of girls a boy \( a \) could marry in some stable matching: this is the set of possible girls for \( a \) . We claim that in the fundamental algorithm no girl in \( S\left( a\right) \) refuses \( a \), so every boy marries his favourite possible girl, and thus the ... | Yes |
Theorem 21 An incomplete system \( \left( {{V}_{1},{V}_{2}, L}\right) \) with \( \left| {V}_{1}\right| = \left| {V}_{2}\right| \) has a stable complete matching iff the associated complete system \( \left( {{V}_{1}^{\prime },{V}_{2}^{\prime },{L}^{\prime }}\right) \) has a stable matching in which the widow marries the... | Proof. Let \( M \) be a complete matching from \( {V}_{1} \) to \( {V}_{2} \), and let \( {M}^{\prime } \) be the complete matching from \( {V}_{1}^{\prime } \) to \( {V}_{2}^{\prime } \) obtained from \( M \) by adding to it the edge \( {wW} \) . To prove the theorem, we shall show that \( M \) is a stable matching in... | Yes |
Theorem 22 No matter what the orders of preferences are, there is always an optimal stable admissions scheme. | Proof. For the sake of argument, call the students boys, and replace each college \( {A}_{i} \) by \( {n}_{i} \) girls, say \( {A}_{i}^{\left( 1\right) },{A}_{i}^{\left( 2\right) },\ldots ,{A}_{i}^{\left( {n}_{i}\right) } \), with each \( {A}_{i}^{\left( j\right) } \) having the same order of preferences among the boys... | Yes |
Theorem 1 For \( g \geq 3 \) and \( \delta \geq 3 \) put\n\n\[ \n{n}_{0}\left( {g,\delta }\right) = \left\{ \begin{array}{ll} 1 + \frac{\delta }{\delta - 2}\left\{ {{\left( \delta - 1\right) }^{\left( {g - 1}\right) /2} - 1}\right\} & \text{ if }g\text{ is odd,} \\ \frac{2}{\delta - 2}\left\{ {{\left( \delta - 1\right)... | Proof. Suppose first that \( g \) is odd, say \( g = {2d} + 1, d \geq 1 \) . Pick a vertex \( x \) . There is no vertex \( z \) for which \( g \) contains two distinct \( z - x \) paths of length at most \( d \), since otherwise \( G \) has a cycle of length at most \( {2d} \) . Consequently, there are at least \( \del... | Yes |
Theorem 2 Let \( G \) be a connected graph of order \( n \geq 3 \) such that for any two non-adjacent vertices \( x \) and \( y \) we have\n\n\[ d\left( x\right) + d\left( y\right) \geq k \]\n\nIf \( k = n \) then \( G \) is Hamiltonian, and if \( k < n \) then \( G \) contains a path of length \( k \) and a cycle of l... | Proof. Assume that \( G \) is not Hamiltonian and let \( P = {x}_{1}{x}_{2}\cdots {x}_{\ell } \) be a longest path in \( G \) . The maximality of \( P \) implies that the neighbours of \( {x}_{1} \) and \( {x}_{\ell } \) are vertices of \( P \) . As \( G \) does not contain a cycle of length \( \ell ,{x}_{1} \) is not ... | Yes |
Theorem 3 Let \( G \) be a graph of order \( n \) without a path of length \( k\left( { \geq 1}\right) \) . Then\n\n\[ e\left( G\right) \leq \frac{k - 1}{2}n \] | Proof. We fix \( k \) and apply induction on \( n \) . The assertion is clearly true if \( n \leq k \) . Assume now that \( n > k \) and the assertion holds for smaller values of \( n \) .\n\nIf \( G \) is disconnected, then the induction hypothesis implies the result. Now, if \( G \) is connected, then it contains no ... | Yes |
Theorem 4 Let \( k \geq 2 \) and let \( G \) be a graph of order \( n \) in which every cycle has length at most \( k \) . Then\n\n\[ e\left( G\right) \leq \frac{k}{2}\left( {n - 1}\right) \]\n\nA graph is extremal iff it is connected and all its blocks are complete graphs of order \( k \) . | The proof of this result is somewhat more involved than that of Theorem 3. Since a convenient way of presenting it uses \ | No |
Theorem 5 Let \( G \) be a graph with \( n \) vertices and at least \( {t}_{r}\left( n\right) \) edges, and let \( x \) be a vertex of maximal degree, say, \( d\left( x\right) = n - k = \Delta \left( G\right) \) . Set \( W = \Gamma \left( x\right) \) , \( U = V\left( G\right) \smallsetminus W \) and \( H = G\left\lbrac... | Proof. As we noted above, \( k \leq \lfloor n/r\rfloor \) . Assume that \( e\left( H\right) \leq {t}_{r - 1}\left( {n - k}\right) \) . Then\n\n\[ \n{t}_{r}\left( n\right) \leq e\left( G\right) = e\left( H\right) + \frac{1}{2}\mathop{\sum }\limits_{{u \in U}}d\left( u\right) + \frac{1}{2}e\left( {U, W}\right) \n\]\n\n\[... | Yes |
Theorem 6 Let \( G \) be a graph with \( n \) vertices and at least \( {t}_{r}\left( n\right) \) edges. Consider the following simple algorithm for finding a complete subgraph of order \( r + 1 \) . Pick a vertex \( {x}_{1} \) of maximal degree in \( {G}_{1} = G \), then a vertex \( {x}_{2} \) of maximal degree in the ... | Proof. We apply induction on \( r \), noting that for \( r = 1 \) there is nothing to prove. Set \( n - k = d\left( {x}_{1}\right) = \Delta \left( G\right) \) . If \( e\left( {G}_{2}\right) > {t}_{r - 1}\left( {n - k}\right) \), then we are done by the induction hypothesis, since \( {G}_{2} \) cannot be isomorphic to \... | Yes |
Theorem 7 Let \( G \) be a graph with vertex set \( V \) that does not contain \( {K}_{r + 1} \), a complete graph of order \( r \) . Then there is an \( r \) -partite graph \( H \) with vertex set \( V \) such that for every vertex \( z \in V \) we have\n\n\[ \n{d}_{G}\left( z\right) \leq {d}_{H}\left( z\right) \n\]\n... | Proof. We shall apply induction on \( r \) . For \( r = 1 \) there is nothing to prove, since \( G \) is the empty graph \( \overline{{K}_{n}} \), which is 1-partite. Assume now that \( r \geq 2 \) and the assertion holds for smaller values of \( r \) .\n\nPick a vertex \( x \in V \) for which \( {d}_{G}\left( x\right)... | Yes |
Theorem 8 For \( r, n \geq 2 \) we have \( \operatorname{ex}\left( {n;{K}_{r + 1}}\right) = {t}_{r}\left( n\right) \) and \( \operatorname{EX}\left( {n;{K}_{r + 1}}\right) = \left\{ {{T}_{r}\left( n\right) }\right\} \) . In words, every graph of order \( n \) with more than \( {t}_{r}\left( n\right) \) edges contains a... | Proof. The theorem is contained in Theorem 6, and it is also an immediate consequence of Theorem 7, since \( {T}_{r}\left( n\right) \) is the unique \( r \) -partite graph of order \( n \) and maximal size.\n\nNevertheless, let us give two more proofs of the theorem itself, based again on the properties of \( {T}_{r}\l... | Yes |
Lemma 9 Let \( m, n, s, t, k, r \) be non-negative integers, \( 2 \leq s \leq m,2 \leq t \leq n \) , \( 0 \leq r < m \), and let \( G = {G}_{2}\left( {m, n}\right) \) be an \( m \) by \( n \) bipartite graph of size \( z = {my} = {km} + r \) without a \( {K}_{s, t} \) subgraph having \( s \) vertices in the first class... | Proof. Denote by \( {V}_{1} \) and \( {V}_{2} \) the vertex classes of \( G \) . We shall say that a \( t \) -set (i.e., a set with \( t \) elements) \( T \) of \( {V}_{2} \) is covered by a vertex \( x \in {V}_{1} \) if \( x \) is joined to every vertex in \( T \) . The number of \( t \) -sets covered by a vertex \( x... | Yes |
Theorem 10 For all natural numbers \( m, n, s \) and \( t \) we have\n\n\[ z\left( {m, n;s, t}\right) \leq {\left( s - 1\right) }^{1/t}\left( {n - t + 1}\right) {m}^{1 - 1/t} + \left( {t - 1}\right) m. \] | Proof. Let \( G = {G}_{2}\left( {m, n}\right) \) be an extremal graph for the function \( z\left( {m, n;s, t}\right) = \) \( {my} \) without a \( K\left( {s, t}\right) \) subgraph. As \( y \geq n \), inequality (2) implies\n\n\[ {\left( y - \left( t - 1\right) \right) }^{t} \leq \left( {s - 1}\right) {\left( n - \left(... | No |
Theorem 11 Let \( n, s, t, k \) and \( r \) be non-negative integers, and let \( G \) be a graph of order \( z = {ny}/2 = \frac{1}{2}\left( {{kn} + r}\right) \), containing no \( {K}_{s, t} \) . Then\n\n\[ \n n\left( \begin{array}{l} y \\ t \end{array}\right) \leq \left( {n - r}\right) \left( \begin{array}{l} k \\ t \e... | Proof. As in Lemma 9, let us say that a \( t \) -set of the vertices is covered by a vertex \( x \) if \( x \) is joined to every vertex of the \( t \) -set. Since \( G \) does not contain a \( {K}_{s, t} \), every \( t \) -set is covered by at most \( s - 1 \) vertices. Therefore, if \( G \) has degree sequence \( {\l... | Yes |
Theorem 12 For \( n \geq 1 \), we have\n\n\[ z\left( {n, n;2,2}\right) \leq \frac{1}{2}n\left\{ {1 + {\left( 4n - 3\right) }^{1/2}}\right\} \]\n\nand equality holds for infinitely many values of \( n \) . Furthermore,\n\n\[ \operatorname{ex}\left( {n,{C}_{4}}\right) \leq \frac{n}{4}\left( {1 + \sqrt{{4n} - 3}}\right) \... | Proof. Since \( \operatorname{lex}\left( {n,{K}_{s, t}}\right) \leq z\left( {n, n;s, t}\right) \), the second inequality is immediate from the first. Moreover, the first inequality is just the case \( s = 2 \) of (4). In fact, the proof of Lemma 9 tells us a considerable amount about the graphs \( G \) for which equali... | Yes |
Lemma 13 A graph \( G \) is Hamiltonian iff \( {C}_{n}\left( G\right) \) is, and \( G \) has a Hamilton path iff \( {C}_{n - 1}\left( G\right) \) does. | Null | No |
Lemma 14 Let \( G \) be a graph with vertex set \( V\left( G\right) = \left\{ {{x}_{1},{x}_{2},\ldots ,{x}_{n}}\right\} \), whose \( k \) -closure \( {C}_{k}\left( G\right) \) contains at most \( t \leq n - 2 \) vertices of degree \( n - 1 \) . Then there are indices \( i, j,1 \leq i < j \leq n \), such that \( {x}_{i}... | Proof. The graph \( H = {C}_{k}\left( G\right) \) is not complete so, we can define two indices \( i \) and \( j \) as follows:\n\n\[ j = \max \left\{ {\ell : {d}_{H}\left( {x}_{\ell }\right) \neq n - 1}\right\} \]\n\n\[ i = \max \left\{ {\ell : {x}_{\ell }{x}_{j} \notin E\left( H\right) }\right\} \]\n\nThen \( {x}_{i}... | Yes |
Theorem 15 Let \( G \) be a graph with vertex set \( V\left( G\right) = \left\{ {{x}_{1},{x}_{2},\ldots ,{x}_{n}}\right\}, n \geq 3 \) . Let \( \varepsilon = 0 \) or 1 and suppose there are no indices \( i, j,1 \leq i < j \leq n \), such that \( {x}_{i}{x}_{j} \notin E\left( G\right) \) and\n\n\[ j \geq n - i + \vareps... | Null | No |
Corollary 16 Let \( G \) be a graph with degree sequence \( {d}_{1} \leq {d}_{2} \leq \cdots \leq {d}_{n} \) , \( n \geq 3 \), and let \( \varepsilon = 0 \) or 1 . Suppose\n\n\[ \n{d}_{n - k + \varepsilon } \geq n - k\\text{ whenever }{d}_{k} \leq k - \varepsilon < \\frac{1}{2}\\left( {n - \varepsilon }\\right) .\n\]\n... | Null | No |
Theorem 18 Let \( W \) be the set of vertices of even degree in a graph \( G \) and let \( {x}_{0} \) be a vertex of \( G \) . Then there is an even number of longest \( {x}_{0} \) -paths ending in \( W \) . | Proof. Let \( H \) be the graph whose vertex set is the set \( \sum \) of longest \( {x}_{0} \) -paths in \( G \) , in which \( {P}_{1} \in \sum \) is joined to \( {P}_{2} \in \sum \) if \( {P}_{2} \) is a simple transform of \( {P}_{1} \) . Since the degree of \( P = {x}_{0}{x}_{1}\cdots {x}_{k} \in \sum \) in \( H \)... | Yes |
Theorem 19 Let \( G \) be a graph in which every vertex has odd degree. Then every edge of \( G \) is contained in an even number of Hamilton cycles. | Proof. Let \( {x}_{0}y \in E\left( G\right) \) . Then in \( {G}^{\prime } = G - {x}_{0}y \) only \( {x}_{0} \) and \( y \) have even degree, so in \( {G}^{\prime } \) there is an even number of longest \( {x}_{0} \) -paths that end in \( y \) . Thus either \( G \) has no Hamilton cycle that contains \( {x}_{0}y \) or i... | Yes |
Lemma 21 Let \( c,\varepsilon > 0 \) . If \( n \) is sufficiently large, say \( n > 3/\varepsilon \), then every graph of order \( n \) and size at least \( \left( {c + \varepsilon }\right) \left( \begin{array}{l} n \\ 2 \end{array}\right) \) contains a subgraph \( H \) with \( \delta \left( H\right) \geq c\left| H\rig... | Proof. Let \( G \) be a graph of order \( n > 3/\varepsilon \) and size \( e\left( G\right) \geq \left( {c + \varepsilon }\right) \left( \begin{array}{l} n \\ 2 \end{array}\right) \) . Note that in this case \( 0 < \varepsilon < \varepsilon + c \leq 1 \) . If the assertion fails then there is a sequence of graphs \( {G... | Yes |
Theorem 22 Let \( r \geq 1 \) be an integer and let \( \varepsilon > 0 \) . Then there is an integer \( {n}_{0} = {n}_{0}\left( {r,\varepsilon }\right) \) such that if \( \left| G\right| = n \geq {n}_{0} \) and\n\n\[ e\left( G\right) \geq \left( {1 - \frac{1}{r} + \varepsilon }\right) \left( \begin{array}{l} n \\ 2 \en... | Proof. If \( n > 3/\varepsilon \) then, by Lemma 21, \( G \) has a subgraph \( H \) with \( \left| H\right| = h \geq {\varepsilon }^{1/2}n \) and \( \delta \left( H\right) \geq \left( {1 - \frac{1}{r} + \varepsilon /2}\right) h \) . Hence if \( n \) is sufficiently large then \( H \) contains a \( {K}_{r + 1}\left( t\r... | Yes |
Corollary 23 Let \( F = {K}_{r + 1}\left( t\right) \), where \( r \geq 1 \) and \( t \geq 1 \) . Then the maximal size of a graph of order \( n \) without a \( {K}_{r + 1}\left( t\right) \) is\n\n\[ \operatorname{ex}\left( {n;F}\right) = \left( {1 - \frac{1}{r}}\right) \left( \begin{array}{l} n \\ 2 \end{array}\right) ... | Null | No |
Corollary 24 Let \( {F}_{1},{F}_{2},\ldots ,{F}_{\ell } \) be non-empty graphs. Denote by \( r + 1 \) the minimum of the chromatic numbers of the \( {F}_{i} \), that is, let \( r + 1 \) be the minimal number for which at least one of the \( {F}_{i} \) is contained in an \( F = {K}_{r + 1}\left( t\right) \) for some t. ... | Proof. The Turán graph \( {T}_{r}\left( n\right) \) does not contain any of the \( {F}_{i} \) so, by (1),\n\n\[ \operatorname{ex}\left( {n;{F}_{1},{F}_{2},\ldots ,{F}_{\ell }}\right) \geq e\left( {{T}_{r - 1}\left( n\right) }\right) = {t}_{r - 1}\left( n\right) \leq \left( {1 - \frac{1}{r}}\right) \left( \begin{array}{... | Yes |
Corollary 25 The upper density of an infinite graph \( G \) is \( 1,\frac{1}{2},\frac{2}{3},\frac{3}{4},\ldots \), or 0 . Each of these values is the upper density of some infinite graph. | Proof. Let \( {G}_{r} \) be the complete \( r \) -partite graph with infinitely many vertices in each class. Since the density of \( {K}_{r}\left( t\right) \) tends to \( 1 - \frac{1}{r} \) as \( t \) tends to \( \infty \), the upper density of \( {G}_{r} \) is \( 1 - \frac{1}{r} \), proving the second assertion.\n\nNo... | Yes |
Lemma 26 Suppose that \( X \) and \( Y \) are disjoint sets of vertices of a graph \( G \) , and \( {X}^{ * } \subset X \) and \( {Y}^{ * } \subset Y \) are such that \( \left| {X}^{ * }\right| \geq \left( {1 - \gamma }\right) \left| X\right| > 0 \) and \( \left| {Y}^{ * }\right| \geq \left( {1 - \delta }\right) \left|... | Proof. Note that, rather crudely,\n\n\[ 0 \leq e\left( {X, Y}\right) - e\left( {{X}^{ * },{Y}^{ * }}\right) \leq \left( {\gamma + \delta - {\gamma \delta }}\right) \left| X\right| \left| Y\right| < \left( {\gamma + \delta }\right) \left| X\right| \left| Y\right| ,\]\n\nso\n\n\[ d\left( {X, Y}\right) - d\left( {{X}^{ * ... | Yes |
Lemma 27 Let \( {\left( {d}_{i}\right) }_{i = 1}^{s} \subset \mathbb{R},1 \leq t < s, D = \frac{1}{s}\mathop{\sum }\limits_{{i = 1}}^{s}{d}_{i} \), and \( d = \frac{1}{t}\mathop{\sum }\limits_{{i = 1}}^{t}{d}_{i} \). Then \[ \frac{1}{s}\mathop{\sum }\limits_{{i = 1}}^{s}{d}_{i}^{2} \geq {D}^{2} + \frac{t}{s - t}{\left(... | Proof. With \[ e = \frac{1}{s - t}\mathop{\sum }\limits_{{i = t + 1}}^{s}{d}_{i} = \frac{{sD} - {td}}{s - t} \] the convexity of the function \( {x}^{2} \) implies that \[ \mathop{\sum }\limits_{{i = 1}}^{s}{d}_{i}^{2} = \mathop{\sum }\limits_{{i = 1}}^{t}{d}_{i}^{2} + \mathop{\sum }\limits_{{i = t + 1}}^{s}{d}_{i}^{2}... | Yes |
Lemma 28 Let \( \;G \) be a graph of order \( \;n\; \) with an equitable partition \( \;V = \mathop{\bigcup }\limits_{{i = 0}}^{k}{C}_{i} \) of the vertex set with exceptional class \( {C}_{0} \) and\n\n\[ \n\left| {C}_{1}\right| = \left| {C}_{2}\right| = \cdots = \left| {C}_{k}\right| = c \geq {2}^{{3k} + 1}.\n\]\n\nS... | Proof. For a pair \( \left( {{C}_{i},{C}_{j}}\right) \) that is not \( \varepsilon \) -uniform, let \( {C}_{ij} \subset {C}_{i} \) and \( {C}_{ji} \subset {C}_{j} \) be sets showing that \( \left( {{C}_{i},{C}_{j}}\right) \) is not \( \varepsilon \) -uniform: \( \left| {C}_{ij}\right| \geq \varepsilon \left| {C}_{i}\ri... | Yes |
Theorem 30 Let \( f \geq 2, r \geq 2,0 < \delta < 1/r \) and let \( {V}_{1},{V}_{2},\ldots ,{V}_{r} \) be disjoint subsets of vertices of a graph \( G \) . Suppose \( \left| {V}_{i}\right| \geq {\delta }^{-f} \) for every \( i \), and if \( 1 \leq i < j \leq r \) and \( {W}_{i} \subset {V}_{i},{W}_{j} \subset {V}_{j} \... | Proof. Let us apply induction on \( f \) . For \( f = 2 \) the assertion is trivial, so suppose that \( f \geq 3 \) and the assertion holds for smaller values of \( f \) . We may assume that \( {f}_{1} \geq 1 \) .\n\nFor \( 2 \leq i \leq r \), let \( {R}_{i} \) be the set of vertices in \( {V}_{1} \) joined to fewer th... | Yes |
Theorem 31 Let \( f \geq 2, r \geq 2,0 < \delta < 1/r \), and let \( {V}_{1},\ldots ,{V}_{r} \) be disjoint subsets of vertices of a graph \( G \) . Suppose \( \left| {V}_{i}\right| \geq {\delta }^{-f} \) for every \( i \), and all pairs \( \left( {{V}_{i},{V}_{j}}\right) \) are \( {\delta }^{f} \) -regular, with densi... | Null | No |
Theorem 32 Let \( 0 < \varepsilon < 1 \) and \( 0 < \delta < 1 \) be real numbers, let \( m \geq 2 \) be an integer, and let \( M = {M}^{\prime \prime }\left( {\varepsilon, m}\right) \) be as in Theorem \( {29}^{\prime \prime } \) of the previous section. Let \( G \) be a graph of order \( n \geq M \), and let \( H = G... | Proof. Let \( V\left( G\right) = \mathop{\bigcup }\limits_{{i = 0}}^{k}{C}_{i} \) be the partition guaranteed by Theorem \( {29}^{\prime \prime } \) so that \( \left| {C}_{0}\right| \leq k - 1,\left| {C}_{1}\right| = \cdots = \left| {C}_{k}\right|, m \leq k \leq M \), and \( H \) is the appropriate \( k \) -partite gra... | Yes |
Theorem 33 For every \( \varepsilon > 0 \) and graph \( F \), there is a constant \( {n}_{0} = {n}_{0}\left( {\varepsilon, F}\right) \) with the following property. Let \( G \) be a graph of order \( n \geq {n}_{0} \) that does not contain \( F \) as a subgraph. Then \( G \) contains a set \( {E}^{\prime } \) of less t... | Proof. We may assume that \( r \geq 2,0 < \varepsilon < 1/r \), and \( f = \left| F\right| \geq 3 \) . Let \( \delta = \varepsilon /2 \) and \( m \geq 8/\varepsilon = 4/\delta \) . Let \( M = {M}^{\prime \prime }\left( {{\delta }^{f}, m}\right) \) be given by Szemerédi’s lemma, as in Theorem \( {29}^{\prime \prime } \)... | Yes |
Theorem 35 Let \( r \geq 3 \) and \( s \geq 2 \) be fixed integers, and \( c \) and \( \gamma \) positive constants. Then if \( n \) is sufficiently large and \( G \) is a graph of order \( n \) that contains neither \( {K}_{r} \) nor \( {K}_{s, t} \), where \( t = \lceil {cn}\rceil \), then\n\n\[ e\left( G\right) \leq... | Proof. We may assume that \( 0 < \gamma < 1/2 \) and \( c < \left( {r - 2}\right) /\left( {r - 1}\right) \), since we do know that \( e\left( G\right) \leq {t}_{r - 1}\left( n\right) \leq \frac{r - 2}{2\left( {r - 1}\right) }{n}^{2} \) .\n\nLet \( \delta = \gamma /2, m \geq 4/\delta \), and suppose that \( n \geq {4Ms}... | Yes |
Theorem 36 For every \( \varepsilon > 0 \) and \( \Delta \geq 1 \) there is an \( {n}_{0} = {n}_{0}\left( {\varepsilon ,\Delta }\right) \) such that every graph of order \( n \) and minimal degree at least \( \left( {1 + \varepsilon }\right) n/2 \) contains every tree of order \( n \) and maximal degree at most \( \Del... | Null | No |
Theorem 1 Let \( k = \mathop{\max }\limits_{H}\delta \left( H\right) \), where the maximum is taken over all induced subgraphs of \( G \) . Then \( \chi \left( G\right) \leq k + 1 \) . | Proof. The graph \( G \) itself has a vertex of degree at most \( k \) ; let \( {x}_{n} \) be such a vertex, and put \( {H}_{n - 1} = G - \left\{ {x}_{n}\right\} \) . By assumption, \( {H}_{n - 1} \) has a vertex of degree at most \( k \) . Let \( {x}_{n - 1} \) be one of them and put \( {H}_{n - 2} = {H}_{n - 1} - \le... | Yes |
Theorem 2 Let \( {H}_{0} \) be an induced subgraph of \( G \) and suppose every subgraph \( H \) satisfying \( {H}_{0} \subset H \subset G, V\left( {H}_{0}\right) \neq V\left( H\right) \), contains a vertex \( x \in V\left( H\right) - V\left( {H}_{0}\right) \) with \( {d}_{H}\left( x\right) \leq k \) . Then \[ \chi \le... | Null | No |
Theorem 3 Let \( G \) be a connected graph with maximal degree \( \Delta \) . Suppose \( G \) is neither a complete graph nor an odd cycle. Then \( \chi \left( G\right) \leq \Delta \) . | Proof. We know already that we may assume without loss of generality that \( G \) is 2-connected and \( \Delta \) -regular. Furthermore, we may assume that \( \Delta \geq 3 \), since a connected 2-regular 3-chromatic graph is an odd cycle.\n\n be a graph with \( n \geq 1 \) vertices, \( m \) edges and \( k \) components. Then\n\n\[ \n{p}_{H}\left( x\right) = \mathop{\sum }\limits_{{i = 0}}^{{n - k}}{\left( -1\right) }^{i}{a}_{i}{x}^{n - i}, \]\n\nwhere \( {a}_{0} = 1,{a}_{1} = m \) and \( {a}_{i} \) is a positive integer for every \( i,... | Proof. We apply induction on \( n + m \) . For \( n + m = 1 \) the assertions are trivial so we pass to the induction step. If \( m = 0 \), we are again done, since in this case \( k = n \) and, as every map \( f : V\left( H\right) \rightarrow \{ 1,2,\ldots, x\} \) is a colouring of \( H \), we have \( {p}_{H}\left( x\... | Yes |
Theorem 5 Let \( H \) be a graph with \( n \) vertices and edge set \( E\left( H\right) = \left\{ {{e}_{1},{e}_{2},\ldots ,{e}_{m}}\right\} \). Call a subset of \( E\left( H\right) \) a broken cycle if it is obtained from the edge set of a cycle by deleting the edge of highest index. Then the chromatic polynomial of \(... | Proof. Let us apply induction on \( m \). For \( m = 0 \) the assertion is trivial, so suppose that \( m \geq 1 \) and the assertion holds for smaller values of \( m \). Let \( {e}_{1} = {ab} \) and, as before, set \( G = H - {ab} \), so that \( {G}^{\prime } = G + {ab} = H \) and \( {G}^{\prime \prime } = G/{ab} \) sa... | Yes |
Corollary 6 Let \( H \) be a graph with \( n \) vertices, \( m \) edges, girth \( g \) and chromatic polynomial\n\n\[ \n{p}_{H}\left( x\right) = \mathop{\sum }\limits_{{i = 0}}^{n}{\left( -1\right) }^{i}{a}_{i}{x}^{n - i}.\n\]\n\nThen \( {a}_{i} = \left( \begin{matrix} m \\ i \end{matrix}\right) \) for \( i \leq g - 2 ... | Null | No |
Theorem 9 The chromatic number of a graph \( G \) drawn on a closed surface of Euler characteristic \( \chi \leq 1 \) is at most \[ h\left( \chi \right) = \lfloor \left( {7 + \sqrt{{49} - {24\chi }}}\right) /2\rfloor . \] | Proof. Let \( k \) be the chromatic number of \( G \) . We may and shall assume that \( G \) is a minimal graph of chromatic number \( k \) ; otherwise, we may replace it by a subgraph. But then \( \delta \left( G\right) \geq k - 1 \), so all we need is that if, for \( h = h\left( \chi \right), G \) has \( n \geq h + 1... | Yes |
Theorem 10 Let \( \chi \leq 0, h = h\left( \chi \right) = \lfloor \left( {7 + \sqrt{{49} - {24\chi }}}\right) /2\rfloor \), and let \( G \) be a minimal h-chromatic graph drawn on a surface of Euler characteristic \( \chi \) . If \( \chi \neq - 1, - 2 \) or -7 then \( G = {K}_{h} \) . | Proof. All we shall use is inequality (8): a graph of order \( n \) drawn on a surface of Euler characteristic \( \chi \) has at most \( 3\left( {n - \chi }\right) \) edges.\n\nSuppose \( G \neq {K}_{h} \) . Then \( n \geq h + 2 \) . Furthermore, if \( n = h + 2 \) then, as claimed by Exercise 38,\n\n\[ e\left( G\right... | Yes |
Theorem 11 The torus, the projective plane and the Klein bottle have chromatic numbers \( s\left( {S}_{1}\right) = 7, s\left( {N}_{1}\right) = 6 \) and \( s\left( {N}_{2}\right) = 6 \) . | Proof. The Euler characteristics of these surfaces are \( \chi \left( {N}_{1}\right) = 1 \) and \( \chi \left( {S}_{1}\right) = \) \( \chi \left( {N}_{2}\right) = 0 \), therefore Theorem 9 implies that \( s\left( {N}_{1}\right) \leq 6 \) and \( s\left( {S}_{1}\right), s\left( {N}_{2}\right) \leq 7 \) . Fig. V. 6 shows ... | Yes |
Theorem 12 Let \( G \) be a near-triangulation with outer cycle \( C = {x}_{1}{x}_{2}\cdots {x}_{k} \) , and for each \( x \in V\left( G\right) \) let \( L\left( x\right) \) be a list of colours assigned to \( x \), such that \( L\left( {x}_{1}\right) = \{ 1\}, L\left( {x}_{2}\right) = \{ 2\} ,\left| {L\left( x\right) ... | Proof. Let us apply induction on the order of \( G \) . For \( \left| G\right| = 3 \) the assertion is trivial, so suppose that \( \left| G\right| > 3 \) and the assertion holds for graphs of order less than \( \left| G\right| \) . We shall distinguish two cases, according to whether \( C \) contains a ’diagonal’ from ... | Yes |
Theorem 13 Let \( G \) be a bipartite graph with total function \( {t}_{G} \) given by a certain assignment of preferences. Then \( G \) is \( \left( {{t}_{G} + 1}\right) \) -choosable. | Proof. We apply induction on the size of \( G \) . If \( E\left( G\right) = \varnothing \), there is nothing to prove, so suppose \( E\left( G\right) \neq \varnothing \) and the assertion holds for graphs of smaller size.\n\nLet us fix an assignment of preferences for \( G \) . For each edge \( e \in E\left( G\right) \... | Yes |
Theorem 14 The list-chromatic index of a bipartite graph equals its chromatic index. | Proof. Let \( G \) be a bipartite graph with bipartition \( \left( {{V}_{1},{V}_{2}}\right) \), and let \( \lambda : E\left( G\right) \rightarrow \) \( \left\lbrack k\right\rbrack \) be an edge-colouring of \( G \), where \( k \) is the chromatic index of \( G \) . Define preferences on \( G \) as follows: let \( a \in... | Yes |
Theorem 16 Let \( G \) be a bipartite graph with line graph \( H = L\left( G\right) \) . Then \( H \) and \( \overline{H} \) are perfect. | Proof. Once again, all we have to prove is that \( \chi \left( H\right) = \omega \left( H\right) \) and \( \chi \left( \bar{H}\right) = \omega \left( \bar{H}\right) \) . Clearly, \( \omega \left( H\right) = \Delta \left( G\right) \) and \( \chi \left( H\right) = {\chi }^{\prime }\left( G\right) \) . But as \( G \) is b... | Yes |
Theorem 17 Comparability graphs and their complements are perfect. | Proof. Once again, it suffices to show that if \( P \) is a partially ordered set then for \( H = C\left( P\right) \) we have \( \chi \left( H\right) = \omega \left( H\right) \) and \( \chi \left( \bar{H}\right) = \omega \left( \bar{H}\right) \) .\n\nTo see the first equality, for \( x \in P \) let \( r\left( x\right) ... | Yes |
A necessary and sufficient condition for a graph \( G \) to be perfect is that for every induced subgraph \( H \subset G \) there is an independent set of vertices, \( I \) , such that\n\n\[ \omega \left( {H - I}\right) < \omega \left( H\right) \]\n\nThat is, a graph is perfect iff every induced subgraph \( H \) has an... | Proof. The necessity holds with plenty to spare. Indeed, let \( H \) be a graph with \( k = \chi \left( H\right) = \omega \left( H\right) \), and let \( I \) be a colour class of a \( k \) -colouring of \( H \) . Then \( \omega \left( {H - I}\right) \leq \chi \left( {H - I}\right) = \chi \left( H\right) - 1 < \omega \l... | Yes |
Theorem 19 A graph obtained from a perfect graph by replacing its vertices by perfect graphs is perfect. | Proof. As we may replace the vertices one by one, it suffices to prove that if a vertex \( x \) of a perfect graph \( G \) is replaced by a perfect graph \( {G}_{x} \) then the resulting graph \( {G}^{ * } \) is perfect. Furthermore, since every induced subgraph of \( {G}^{ * } \) is of precisely the same form (obtaine... | Yes |
Theorem 1 The function \( R\left( {s, t}\right) \) is finite for all \( s, t \geq 2 \) . If \( s > 2 \) and \( t > 2 \) then\n\n\[ R\left( {s, t}\right) \leq R\left( {s - 1, t}\right) + R\left( {s, t - 1}\right) \]\n\n(1)\n\nand\n\n\[ R\left( {s, t}\right) \leq \left( \begin{matrix} s + t - 2 \\ s - 1 \end{matrix}\righ... | Proof. As we shall prove (1) and (2), it will follow that \( R\left( {s, t}\right) \) is finite.\n\n(i) When proving (1) we may assume that \( R\left( {s - 1, t}\right) \) and \( R\left( {s, t - 1}\right) \) are finite. Let \( n = R\left( {s - 1, t}\right) + R\left( {s, t - 1}\right) \) and consider a colouring of the ... | Yes |
Theorem 2 Let \( 1 < r < \min \{ s, t\} \) . Then \( {R}^{\left( r\right) }\left( {s, t}\right) \) is finite and\n\n\[ \n{R}^{\left( r\right) }\left( {s, t}\right) \leq {R}^{\left( r - 1\right) }\left( {{R}^{\left( r\right) }\left( {s - 1, t}\right) ,{R}^{\left( r\right) }\left( {s, t - 1}\right) }\right) + 1.\n\] | Proof. Both assertions follow immediately if we prove the inequality under the assumption that \( {R}^{\left( r - 1\right) }\left( {u, v}\right) \) is finite for all \( u, v \), and both \( {R}^{\left( r\right) }\left( {s - 1, t}\right) \) and \( {R}^{\left( r\right) }\left( {s, t - 1}\right) \) are also finite.\n\nLet... | Yes |
Theorem 3 For \( k,\ell \geq 2 \), every non-degenerate set of \( \left( \begin{matrix} k + \ell - 4 \\ k - 2 \end{matrix}\right) + 1 \) points contains a \( k \) -cup or an \( \ell \) -cap. | Proof. Let us write \( \phi \left( {k,\ell }\right) \) for the binomial coefficient \( \left( \begin{matrix} k + \ell - 4 \\ k - 2 \end{matrix}\right) \). (i) We shall prove by induction on \( k + \ell \) that every non-degenerate set of \( \phi \left( {k,\ell }\right) + 1 \) points contains a \( k \) -cup or an \( \el... | Yes |
Theorem 4 Let \( 1 \leq r < \infty \) and let \( c : {A}^{\left( r\right) } \rightarrow \left\lbrack k\right\rbrack = \{ 1,2,\ldots, k\} \) be a \( k \) - colouring of the r-tuples of an infinite set \( A \) . Then \( A \) contains a monochromatic infinite set. | Proof. We apply induction on \( r \) . Note that the result is trivial for \( r = 1 \), so we may assume that \( r > 1 \) and the theorem holds for smaller values of \( r \) .\n\nPut \( {A}_{0} = A \) and pick an element \( {x}_{1} \in {A}_{0} \) . As in the proof of Theorem 2, define a a colouring \( {c}_{1} : {B}_{1}... | Yes |
Theorem 5 For each \( r \in \mathbb{N} \), colour the set \( {\mathbb{N}}^{\left( r\right) } \) of \( r \) -tuples of \( \mathbb{N} \) with \( {k}_{r} \) colours, where \( {k}_{r} \in \mathbb{N} \) . Then there is an infinite set \( M \subset \mathbb{N} \) such that for every \( r \) any two \( r \) -tuples of \( M \) ... | Proof. Put \( {M}_{0} = \mathbb{N} \) . Having chosen infinite sets \( {M}_{0} \supset \cdots \supset {M}_{r - 1} \), let \( {M}_{r} \) be an infinite subset of \( {M}_{r - 1} \) such that all the \( r \) -tuples of \( {M}_{r} \) have the same colour. This way we obtain an infinite nested sequence of infinite sets: \( ... | Yes |
Theorem 6 Let \( r \) and \( k \) be natural numbers, and for every \( n \geq 1 \), let \( {\mathcal{C}}_{n} \) be a non-empty set of \( k \) -colourings of \( {\left\lbrack n\right\rbrack }^{\left( r\right) } \) such that if \( n < m \) and \( {c}_{m} \in {\mathcal{C}}_{m} \) then the restriction \( {c}_{m}^{\left( n\... | Proof. For \( m > n \), write \( {\mathcal{C}}_{n, m} \) for the set of colourings \( {\left\lbrack n\right\rbrack }^{\left( r\right) } \rightarrow \left\lbrack k\right\rbrack \) that are restrictions of colourings in \( {\mathcal{C}}_{m} \) . Then \( {\mathcal{C}}_{n, m + 1} \subset {\mathcal{C}}_{n, m} \subset {\math... | Yes |
Theorem 7 Let \( r, k \) and \( s \geq 2 \) . If \( n \) is sufficiently large then for every \( k \) -colouring of \( {\left\lbrack n\right\rbrack }^{\left( r\right) } \) there is a monochromatic set \( S \subset \left\lbrack n\right\rbrack \) such that\n\n\[ \left| S\right| \geq \max \{ s,\min S\} \text{.} \] | Proof. Suppose that there is no such \( n \), that is, for every \( n \) there is a colouring \( {\left\lbrack n\right\rbrack }^{\left( r\right) } \rightarrow \left\lbrack k\right\rbrack \) without an appropriate monochromatic set. Let \( {\mathcal{C}}_{n} \) be the set of all such colourings. Then \( {\mathcal{C}}_{n}... | Yes |
Theorem 10 For \( \ell \geq 1 \) and \( p \geq 2 \) we have\n\n\[ r\left( {\ell {K}_{2},{K}_{p}}\right) = 2\ell + p - 2. \] | Proof. The graph \( {K}_{2\ell - 1} \cup {E}_{p - 2} \) does not contain \( \ell \) independent edges, and its complement, \( {E}_{2\ell - 1} + {K}_{p - 2} \), does not contain a complete graph of order \( p \) . Hence \( r\left( {\ell {K}_{2},{K}_{p}}\right) \geq 2\ell + p - 2 \) .\n\nOn the other hand, let \( G \) be... | Yes |
Theorem 11 For all nonempty graphs \( {H}_{1} \) and \( {H}_{2} \) we have\n\n\[ r\left( {{H}_{1},{H}_{2}}\right) \geq \left( {\chi \left( {H}_{1}\right) - 1}\right) \left( {c\left( {H}_{2}\right) - 1}\right) + u\left( {H}_{1}\right) . \]\n\nIn particular, if \( {\mathrm{H}}_{2} \) is connected then\n\n\[ r\left( {{H}_... | Proof. Set \( k = \chi \left( {H}_{1}\right), u = u\left( {H}_{1}\right) \) and \( c = c\left( {H}_{2}\right) \) . Trivially, \( r\left( {{H}_{1},{H}_{2}}\right) \geq \) \( \left. {r\left( {{H}_{1},{K}_{2}}\right) = \left| {H}_{1}\right| \geq \chi \left( {H}_{1}\right) u\left( {H}_{1}\right) = {ku}\text{. Hence, if }c ... | Yes |
Theorem 12 Let \( s, t \geq 2 \) . then for every tree \( T \) of order \( t \) we have \( r\left( {{K}_{s}, T}\right) = \) \( \left( {s - 1}\right) \left( {t - 1}\right) + 1 \) . | Proof. From Theorem 10 we know that \( r\left( {{K}_{s}, T}\right) \geq \left( {s - 1}\right) \left( {t - 1}\right) + 1 \) . To prove the reverse inequality, let \( G \) be a graph of order \( n = \left( {s - 1}\right) \left( {t - 1}\right) + 1 \) whose complement does not contain \( {K}_{s} \) . Then \( \chi \left( G\... | Yes |
Theorem 13 For \( \ell \geq 2 \) we have \( r\left( {{F}_{1},{F}_{\ell }}\right) = r\left( {{K}_{3},{F}_{\ell }}\right) = 4\ell + 1 \) . | Proof. We know from Theorem 11 that \( r\left( {{K}_{3},{F}_{\ell }}\right) \geq 2\left( {\left| {F}_{\ell }\right| - 1}\right) + 1 = 4\ell + 1 \) . To prove the reverse inequality, suppose that the inequality is false; that is, there is a triangle-free graph \( G \) of order \( n = 4\ell + 1 \) whose complement does n... | Yes |
Lemma 14 For all graphs \( G,{H}_{1} \) and \( {H}_{2} \) we have \( r\left( {G,{H}_{1} \cup {H}_{2}}\right) \leq \) \( \max \left\{ {r\left( {G,{H}_{1}}\right) + \left| {H}_{2}\right|, r\left( {G,{H}_{2}}\right) }\right\} \) . In particular, \( r\left( {s{H}_{1},{H}_{2}}\right) \leq r\left( {{H}_{1},{H}_{2}}\right) + ... | Proof. Let \( n = \max \left\{ {r\left( {G,{H}_{1}}\right) + \left| {H}_{2}\right|, r\left( {G,{H}_{2}}\right) }\right\} \), and suppose that we are given a red-blue colouring of \( {K}_{n} \) without a red \( G \) . Then \( n \geq r\left( {G,{H}_{2}}\right) \) implies that there is a blue \( {H}_{2} \) . Remove it. Si... | Yes |
Theorem 15 If \( s \geq t \geq 1 \) then\n\n\[ r\left( {s{K}_{2}, t{K}_{2}}\right) = {2s} + t - 1. \] | Proof. The graph \( G = {K}_{{2s} - 1} \cup {E}_{t - 1} \) does not contain \( s \) independent edges and \( \bar{G} = {E}_{{2s} - 1} + {K}_{t - 1} \) does not contain \( t \) independent edges. Hence \( r\left( {s{K}_{2}, t{K}_{2}}\right) \geq \) \( {2s} + t - 1 \) .\n\nTrivially (or, by Theorem 10), \( r\left( {s{K}_... | Yes |
Theorem 16 If \( s \geq t \geq 1 \) and \( s \geq 2 \) then \( r\left( {s{K}_{3}, t{K}_{3}}\right) = {3s} + {2t} \) . | Proof. Let \( G = {K}_{{3s} - 1} \cup \left( {{K}_{1} + {E}_{{2t} - 1}}\right) \) . Then \( G \) does not contain \( s \) independent triangles and \( \bar{G} = {E}_{{3s} - 1} + \left( {{K}_{1} \cup {K}_{{2t} - 1}}\right) \) does not contain \( t \) independent triangles. Hence \( r\left( {s{K}_{3}, t{K}_{3}}\right) \)... | Yes |
Theorem 17 If \( s \geq t \geq 1 \) then\n\n\[ \n{ps} + \left( {q - 1}\right) t - 1 \leq r\left( {s{K}_{p}, t{K}_{q}}\right) \leq {ps} + \left( {q - 1}\right) t + C.\n\] | Proof. The graph \( {K}_{{ps} - 1} \cup {E}_{\left( {q - 1}\right) t - 1} \) shows the first inequality. As in the proofs of the previous theorems, we fix \( s - t \) and apply induction on \( t \) . By Lemma 14 we have\n\n\[ \nr\left( {s{K}_{p}, t{K}_{q}}\right) \leq \left( {s - t}\right) p + r\left( {t{K}_{p}, t{K}_{... | Yes |
Theorem 18 For every \( d \geq 1 \) there is a constant \( c = c\left( d\right) \) such that if \( \Delta \left( H\right) \leq d \) then \( r\left( {H, H}\right) \leq c\left| H\right| \) . | Null | No |
Theorem 19 For every graph \( H \) and integer \( k \geq 1 \), there is a graph \( G \) with \( \omega \left( G\right) = \omega \left( H\right) \) such that every \( k \) -colouring of the edges of \( G \) contains a monochromatic induced subgraph isomorphic to \( H \) . | Null | No |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.