1,511 research outputs found

    Constrained Ramsey Numbers

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    For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum n such that every edge coloring of the complete graph on n vertices, with any number of colors, has a monochromatic subgraph isomorphic to S or a rainbow (all edges differently colored) subgraph isomorphic to T. The Erdos-Rado Canonical Ramsey Theorem implies that f(S, T) exists if and only if S is a star or T is acyclic, and much work has been done to determine the rate of growth of f(S, T) for various types of parameters. When S and T are both trees having s and t edges respectively, Jamison, Jiang, and Ling showed that f(S, T) <= O(st^2) and conjectured that it is always at most O(st). They also mentioned that one of the most interesting open special cases is when T is a path. In this work, we study this case and show that f(S, P_t) = O(st log t), which differs only by a logarithmic factor from the conjecture. This substantially improves the previous bounds for most values of s and t.Comment: 12 pages; minor revision

    Generalizations of the Tree Packing Conjecture

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    The Gy\'arf\'as tree packing conjecture asserts that any set of trees with 2,3,...,k2,3, ..., k vertices has an (edge-disjoint) packing into the complete graph on kk vertices. Gy\'arf\'as and Lehel proved that the conjecture holds in some special cases. We address the problem of packing trees into kk-chromatic graphs. In particular, we prove that if all but three of the trees are stars then they have a packing into any kk-chromatic graph. We also consider several other generalizations of the conjecture

    Red-blue clique partitions and (1-1)-transversals

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    Motivated by the problem of Gallai on (1−1)(1-1)-transversals of 22-intervals, it was proved by the authors in 1969 that if the edges of a complete graph KK are colored with red and blue (both colors can appear on an edge) so that there is no monochromatic induced C4C_4 and C5C_5 then the vertices of KK can be partitioned into a red and a blue clique. Aharoni, Berger, Chudnovsky and Ziani recently strengthened this by showing that it is enough to assume that there is no induced monochromatic C4C_4 and there is no induced C5C_5 in {\em one of the colors}. Here this is strengthened further, it is enough to assume that there is no monochromatic induced C4C_4 and there is no K5K_5 on which both color classes induce a C5C_5. We also answer a question of Kaiser and Rabinovich, giving an example of six 22-convex sets in the plane such that any three intersect but there is no (1−1)(1-1)-transversal for them

    Vertex covers by monochromatic pieces - A survey of results and problems

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    This survey is devoted to problems and results concerning covering the vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles and other objects. It is an expanded version of the talk with the same title at the Seventh Cracow Conference on Graph Theory, held in Rytro in September 14-19, 2014.Comment: Discrete Mathematics, 201
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