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Let G be a bipartite graph, with k|e(G). The zero-sum bipartite Ramsey number B(G, Zk) is the smallest integer t such that in every Zk-coloring of the edges of Kt,t, there is a zero-sum mod k copy of G in Kt,t. In this article we give the first proof that determines B(G, Z2) for all possible bipartite graphs G. In fact, we prove a much more general result from which B(G, Z2) can be deduced: Let G be a (not necessarily connected) bipartite graph, which can be embedded in Kn,n, and let F be a field. A function f : E(Kn,n) → F is called G-stable if every copy of G in Kn,n has the same weight (the weight of a copy is the sum of the values of f on its edges). The set of all G-stable functions, denoted by U(G, Kn,n, F) is a linear space, which is called the Kn,n uniformity space of G over F. We determine U(G, Kn,n, F) and its dimension, for all G, n and F. Utilizing this result in the case F = Z2, we can compute B(G, Z2), for all bipartite graphs G. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 151–166, 1998 相似文献
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The Ramsey number r(H,Kn) is the smallest integer N so that each graph on N vertices that fails to contain H as a subgraph has independence number at least n. It is shown that r(K2,m,Kn)(m−1+o(1))(n/log n)2 and r(C2m,Kn)c(n/log n)m/(m−1) for m fixed and n→∞. Also r(K2,n,Kn)=Θ(n3/log2 n) and
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The study of the CO‐irredundant Ramsey numbers t(n1, ···, nk) is initiated. It is shown that several values and bounds for these numbers may be obtained from the well‐studied generalized graph Ramsey numbers and the values of t(4, 5), t(4, 6) and t(3, 3, m) are calculated. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 258–268, 2000 相似文献
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For given graphs G and H, the Ramsey number R(G,H) is the smallest natural number n such that for every graph F of order n: either F contains G or the complement of F contains H. In this paper we investigate the Ramsey number of a disjoint union of graphs . For any natural integer k, we contain a general upper bound, R(kG,H)?R(G,H)+(k-1)|V(G)|. We also show that if m=2n-4, 2n-8 or 2n-6, then R(kSn,Wm)=R(Sn,Wm)+(k-1)n. Furthermore, if |Gi|>(|Gi|-|Gi+1|)(χ(H)-1) and R(Gi,H)=(χ(H)-1)(|Gi|-1)+1, for each i, then . 相似文献
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Ingo Schiermeyer 《Journal of Graph Theory》2003,44(4):251-260
The cycle‐complete graph Ramsey number r(Cm, Kn) is the smallest integer N such that every graph G of order N contains a cycle Cm on m vertices or has independence number α(G) ≥ n. It has been conjectured by Erd?s, Faudree, Rousseau and Schelp that r(Cm, Kn) = (m ? 1) (n ? 1) + 1 for all m ≥ n ≥ 3 (except r(C3, K3) = 6). This conjecture holds for 3 ≤ n ≤ 5. In this paper we will present a proof for n = 6 and for all n ≥ 7 with m ≥ n2 ? 2n. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 251–260, 2003 相似文献
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Let G be a graph on n vertices in which every induced subgraph on vertices has an independent set of size at least . What is the largest so that every such G must contain an independent set of size at least q? This is one of the several related questions raised by Erd?s and Hajnal. We show that , investigate the more general problem obtained by changing the parameters s and t, and discuss the connection to a related Ramsey‐type problem. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 149–157, 2007 相似文献
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Given two graphs G and H, let f(G,H) denote the minimum integer n such that in every coloring of the edges of Kn, there is either a copy of G with all edges having the same color or a copy of H with all edges having different colors. We show that f(G,H) is finite iff G is a star or H is acyclic. If S and T are trees with s and t edges, respectively, we show that 1+s(t?2)/2≤f(S,T)≤(s?1)(t2+3t). Using constructions from design theory, we establish the exact values, lying near (s?1)(t?1), for f(S,T) when S and T are certain paths or star‐like trees. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 1–16, 2003 相似文献
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For given graphs G1,G2,…,Gk, k≥2, the multicolor Ramsey number, denoted by R(G1,G2,…,Gk), is the smallest integer n such that if we arbitrarily color the edges of a complete graph on n vertices with k colors, there is always a monochromatic copy of Gi colored with i, for some 1≤i≤k. Let Pk (resp. Ck) be the path (resp. cycle) on k vertices. In the paper we consider the value for numbers of type R(Pi,Pk,Cm) for odd m, k≥m≥3 and when i is odd, and when i is even. In addition, we provide the exact values for Ramsey numbers R(P3,Pk,C4) for all integers k≥3. 相似文献
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The Ramsey number R(G1,G2,…,Gk) is the least integer p so that for any k-edge coloring of the complete graph Kp, there is a monochromatic copy of Gi of color i. In this paper, we derive upper bounds of R(G1,G2,…,Gk) for certain graphs Gi. In particular, these bounds show that R(9,9)6588 and R(10,10)23556 improving the previous best bounds of 6625 and 23854. 相似文献
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For every ?>0 and every positive integers Δ and r, there exists C=C(?,Δ,r) such that the Ramsey number, R(H,H) of any r-uniform hypergraph H with maximum degree at most Δ is at most C|V(H)|1+?. 相似文献
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Péter Komjáth 《Journal of Graph Theory》2020,94(4):499-508
For every infinite cardinal λ and 2 ≤ n < ω there is a directed graph D of size λ such that D does not contain directed circuits of length ≤n and if its vertices are colored with <λ colors, then there is a monochromatic directed circuit of length n + 1. For every infinite cardinal λ and finite graph X there is a λ-sized graph Y such that if the vertices of Y are colored with <λ colors, then there is a monochromatic induced copy of X. Further, Y does not contain larger cliques or shorter odd circuits than X. The constructions are using variants of Specker-type graphs. 相似文献
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In this paper we study multipartite Ramsey numbers for odd cycles. We formulate the following conjecture: Let n≥5 be an arbitrary positive odd integer; then, in any two‐coloring of the edges of the complete 5‐partite graph K((n?1)/2, (n?1)/2, (n?1)/2, (n?1)/2, 1) there is a monochromatic Cn, a cycle of length n. This roughly says that the Ramsey number for Cn (i.e. 2n?1 ) will not change (somewhat surprisingly) if four large “holes” are allowed. Note that this would be best possible as the statement is not true if we delete from K2n?1 the edges within a set of size (n+ 1)/2. We prove an approximate version of the above conjecture. © 2009 Wiley Periodicals, Inc. J Graph Theory 61:12‐21, 2009 相似文献
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We prove that there is an absolute constant C>0 so that for every natural n there exists a triangle‐free regular graph with no independent set of size at least \({{C}}\sqrt{{{n}}\log{{n}}}\). © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 244–249, 2010 相似文献
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We show that if G is a Ramsey size‐linear graph and x,y ∈ V (G) then if we add a sufficiently long path between x and y we obtain a new Ramsey size‐linear graph. As a consequence we show that if G is any graph such that every cycle in G contains at least four consecutive vertices of degree 2 then G is Ramsey size‐linear. © 2002 John Wiley & Sons, Inc. J Graph Theory 39: 1–5, 2002 相似文献
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András Gyárfás Miklós Ruszinkó Gábor N. Sárközy Endre Szemerédi 《Journal of Graph Theory》2007,55(2):164-174
In this article, we study the tripartite Ramsey numbers of paths. We show that in any two‐coloring of the edges of the complete tripartite graph K(n, n, n) there is a monochromatic path of length (1 ? o(1))2n. Since R(P2n+1,P2n+1)=3n, this means that the length of the longest monochromatic path is about the same when two‐colorings of K3n and K(n, n, n) are considered. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 164–174, 2007 相似文献