35 research outputs found

    Tur\`an numbers of Multiple Paths and Equibipartite Trees

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    The Tur\'an number of a graph H, ex(n;H), is the maximum number of edges in any graph on n vertices which does not contain H as a subgraph. Let P_l denote a path on l vertices, and kP_l denote k vertex-disjoint copies of P_l. We determine ex(n, kP_3) for n appropriately large, answering in the positive a conjecture of Gorgol. Further, we determine ex (n, kP_l) for arbitrary l, and n appropriately large relative to k and l. We provide some background on the famous Erd\H{o}s-S\'os conjecture, and conditional on its truth we determine ex(n;H) when H is an equibipartite forest, for appropriately large n.Comment: 17 pages, 13 figures; Updated to incorporate referee's suggestions; minor structural change

    Generalized Tur\'an problems for disjoint copies of graphs

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    Given two graphs HH and FF, the maximum possible number of copies of HH in an FF-free graph on nn vertices is denoted by ex(n,H,F)ex(n,H,F). We investigate the function ex(n,H,kF)ex(n,H,kF), where kFkF denotes kk vertex disjoint copies of a fixed graph FF. Our results include cases when FF is a complete graph, cycle or a complete bipartite graph.Comment: 18 pages. There was a wrong statement in the first version, it is corrected no

    On the Turan number of forests

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    The Turan number of a graph H, ex(n,H), is the maximum number of edges in a graph on n vertices which does not have H as a subgraph. We determine the Turan number and find the unique extremal graph for forests consisting of paths when n is sufficiently large. This generalizes a result of Bushaw and Kettle [ Combinatorics, Probability and Computing 20:837--853, 2011]. We also determine the Turan number and extremal graphs for forests consisting of stars of arbitrary order

    Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

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    In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs

    Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

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    In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
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