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1.
The -color bipartite Ramsey number of a bipartite graph is the least integer for which every -edge-colored complete bipartite graph contains a monochromatic copy of . The study of bipartite Ramsey numbers was initiated, over 40 years ago, by Faudree and Schelp and, independently, by Gyárfás and Lehel, who determined the 2-color Ramsey number of paths. In this paper we determine asymptotically the 3-color bipartite Ramsey number of paths and (even) cycles. 相似文献
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Let brk(C4;Kn, n) be the smallest N such that if all edges of KN, N are colored by k + 1 colors, then there is a monochromatic C4 in one of the first k colors or a monochromatic Kn, n in the last color. It is shown that brk(C4;Kn, n) = Θ(n2/log2n) for k?3, and br2(C4;Kn, n)≥c(n n/log2n)2 for large n. The main part of the proof is an algorithm to bound the number of large Kn, n in quasi‐random graphs. © 2010 Wiley Periodicals, Inc. J Graph Theory 67: 47‐54, 2011 相似文献
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Fabrıcio Siqueira Benevides 《Journal of Graph Theory》2012,71(3):293-316
In this article we study multipartite Ramsey numbers for odd cycles. Our main result is the proof that a conjecture of Gyárfás et al. (J Graph Theory 61 (2009), 12–21), holds for graphs with a large enough number of vertices. Precisely, there exists n0 such that if n?n0 is a positive odd integer then any two‐coloring of the edges of the complete five‐partite graph K(n ? 1)/2, (n ? 1)/2, (n ? 1)/2, (n ? 1)/2, 1 contains a monochromatic cycle of length n. © 2011 Wiley Periodicals, Inc. J Graph Theory 相似文献
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David Rolnick 《Discrete Mathematics》2013,313(20):2084-2093
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Arès Méroueh 《Journal of Graph Theory》2019,90(2):172-188
Let be the Ramsey number of an -uniform loose cycle of length versus an -uniform clique of order . Kostochka et al. showed that for each fixed , the order of magnitude of is up to a polylogarithmic factor in . They conjectured that for each we have . We prove that , and more generally for every that . We also prove that for every and , if is odd, which improves upon the result of Collier-Cartaino et al. who proved that for every and we have . 相似文献
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Vites Longani 《Southeast Asian Bulletin of Mathematics》2003,26(4):583-592
Consider a complete bipartite graph K(s, s) with p = 2s points. Let each line of the graph have either red or blue colour. The smallest number p of points such that K(s, s) always contains red K(m, n) or blue K(m, n) is called bipartite Ramsey number denoted by rb(K(m, n), K(m, n)). In this paper, we show that
AMS Subject Classifications (1991): 05C15, 05D10. 相似文献
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David Conlon 《Journal of Graph Theory》2008,58(4):351-356
We consider the following question: how large does n have to be to guarantee that in any two‐coloring of the edges of the complete graph Kn,n there is a monochromatic Kk,k? In the late 1970s, Irving showed that it was sufficient, for k large, that n ≥ 2k ? 1 (k ? 1) ? 1. Here we improve upon this bound, showing that it is sufficient to take where the log is taken to the base 2. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 351–356, 2008 相似文献
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For bipartite graphs , the bipartite Ramsey number is the least positive integer so that any coloring of the edges of with colors will result in a copy of in the th color for some . In this paper, our main focus will be to bound the following numbers: and for all for and for Furthermore, we will also show that these mentioned bounds are generally better than the bounds obtained by using the best known Zarankiewicz-type result. 相似文献
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Size bipartite Ramsey numbers 总被引:1,自引:0,他引:1
Yuqin Sun 《Discrete Mathematics》2009,309(5):1060-1066
Let B, B1 and B2 be bipartite graphs, and let B→(B1,B2) signify that any red-blue edge-coloring of B contains either a red B1 or a blue B2. The size bipartite Ramsey number is defined as the minimum number of edges of a bipartite graph B such that B→(B1,B2). It is shown that is linear on n with m fixed, and is between c1n22n and c2n32n for some positive constants c1 and c2. 相似文献
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P.E. Haxell 《Journal of Combinatorial Theory, Series A》2006,113(1):67-83
Let Cn denote the 3-uniform hypergraph loose cycle, that is the hypergraph with vertices v1,…,vn and edges v1v2v3, v3v4v5, v5v6v7,…,vn-1vnv1. We prove that every red-blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of Cn, where N is asymptotically equal to 5n/4. Moreover this result is (asymptotically) best possible. 相似文献
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Given a graph G and a subgraph H of G, let rb(G,H) be the minimum number r for which any edge-coloring of G with r colors has a rainbow subgraph H. The number rb(G,H) is called the rainbow number of H with respect to G. Denote as mK2 a matching of size m and as Bn,k the set of all the k-regular bipartite graphs with bipartition (X,Y) such that X=Y=n and k≤n. Let k,m,n be given positive integers, where k≥3, m≥2 and n>3(m−1). We show that for every GBn,k, rb(G,mK2)=k(m−2)+2. We also determine the rainbow numbers of matchings in paths and cycles. 相似文献
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《Discrete Mathematics》2022,345(5):112801
Let G and H be simple graphs. The Ramsey number is the minimum integer N such that any red-blue-coloring of edges of contains either a red copy of G or a blue copy of H. Let denote m vertex-disjoint copies of . A lower bound is that . Burr, Erd?s and Spencer proved that this bound is indeed the Ramsey number for , and . In this paper, we show that this bound is the Ramsey number for and . We also show that this bound is the Ramsey number for and . 相似文献
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Michael Tait 《Discrete Mathematics》2018,341(1):104-108
Let denote that any -coloring of contains a monochromatic . The degree Ramsey number of a graph , denoted by , is . We consider degree Ramsey numbers where is a fixed even cycle. Kinnersley, Milans, and West showed that , and Kang and Perarnau showed that . Our main result is that and . Additionally, we substantially improve the lower bound for for general . 相似文献
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For given graphs G and H and an integer k, the Gallai–Ramsey number is defined to be the minimum integer n such that, in any k coloring of the edges of Kn, there exists a subgraph isomorphic to either a rainbow coloring of G or a monochromatic coloring of H. In this work, we consider Gallai–Ramsey numbers for the case when G=K3 and H is a cycle of a fixed length. 相似文献
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Michael Savery 《Journal of Graph Theory》2022,99(1):152-161
A graph is Ramsey for a graph if every colouring of the edges of in two colours contains a monochromatic copy of . Two graphs and are Ramsey equivalent if any graph is Ramsey for if and only if it is Ramsey for . A graph parameter is Ramsey distinguishing if implies that and are not Ramsey equivalent. In this paper we show that the chromatic number is a Ramsey distinguishing parameter. We also extend this to the multicolour case and use a similar idea to find another graph parameter which is Ramsey distinguishing. 相似文献
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We consider the binomial random graph Gp and determine a sharp threshold function for the edge-Ramsey property for all l1,…,lr, where Cl denotes the cycle of length l. As deterministic consequences of our results, we prove the existence of sparse graphs having the above Ramsey property as well as the existence of infinitely many critical graphs with respect to the property above. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11 , 245–276, 1997 相似文献