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《Discrete Mathematics》2020,343(12):112117
Let be an edge-colored graph of order . The minimum color degree of , denoted by , is the largest integer such that for every vertex , there are at least distinct colors on edges incident to . We say that an edge-colored graph is rainbow if all its edges have different colors. In this paper, we consider vertex-disjoint rainbow triangles in edge-colored graphs. Li (2013) showed that if , then contains a rainbow triangle and the lower bound is tight. Motivated by this result, we prove that if and , then contains two vertex-disjoint rainbow triangles. In particular, we conjecture that if , then contains vertex-disjoint rainbow triangles. For any integer , we show that if and , then contains vertex-disjoint rainbow triangles. Moreover, we provide sufficient conditions for the existence of edge-disjoint rainbow triangles. 相似文献
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A Necessary and Sufficient Condition for the Existence of a Heterochromatic Spanning Tree in a Graph
Kazuhiro Suzuki 《Graphs and Combinatorics》2006,22(2):261-269
We prove the following theorem. An edge-colored (not necessary to be proper) connected graph G of order n has a heterochromatic spanning tree if and only if for any r colors (1≤r≤n−2), the removal of all the edges colored with these r colors from G results in a graph having at most r+1 components, where a heterochromatic spanning tree is a spanning tree whose edges have distinct colors. 相似文献
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We prove that for every fixed k and ? ≥ 5 and for sufficiently large n, every edge coloring of the hypercube Qn with k colors contains a monochromatic cycle of length 2 ?. This answers an open question of Chung. Our techniques provide also a characterization of all subgraphs H of the hypercube which are Ramsey, that is, have the property that for every k, any k‐edge coloring of a sufficiently large Qn contains a monochromatic copy of H. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 196–208, 2006 相似文献
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关于Ramsey数下界的部分结果 总被引:3,自引:1,他引:2
刘富贵 《数学的实践与认识》2002,32(1):97-99
本文得到 Ramsey数下界的一个计算公式 :R( l,s+ t-2 )≥ R( l,s) + R( l,t) -1 ,(式中 l、s、t≥ 3) .用此公式算得的 Ramsey数的下界比用其它公式算得的下界好 . 相似文献
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Lingsheng Shi 《Journal of Graph Theory》2005,50(3):175-185
The Ramsey number R(G1,G2) of two graphs G1 and G2 is the least integer p so that either a graph G of order p contains a copy of G1 or its complement Gc contains a copy of G2. In 1973, Burr and Erd?s offered a total of $25 for settling the conjecture that there is a constant c = c(d) so that R(G,G)≤ c|V(G)| for all d‐degenerate graphs G, i.e., the Ramsey numbers grow linearly for d‐degenerate graphs. We show in this paper that the Ramsey numbers grow linearly for degenerate graphs versus some sparser graphs, arrangeable graphs, and crowns for example. This implies that the Ramsey numbers grow linearly for degenerate graphs versus graphs with bounded maximum degree, planar graphs, or graphs without containing any topological minor of a fixed clique, etc. © 2005 Wiley Periodicals, Inc. J Graph Theory 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