280 research outputs found

    Embedding large subgraphs into dense graphs

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    What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac's theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect matchings are generalized by perfect F-packings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F. It is unlikely that there is a characterization of all graphs G which contain a perfect F-packing, so as in the case of Dirac's theorem it makes sense to study conditions on the minimum degree of G which guarantee a perfect F-packing. The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy and Szemeredi have proved to be powerful tools in attacking such problems and quite recently, several long-standing problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on F-packings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved

    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

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

    Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

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

    Ramsey numbers for partially-ordered sets

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    We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of partially-ordered sets to form host graphs for Ramsey problems. We explore connections to well studied Tur\'an-type problems in partially-ordered sets, particularly those in the Boolean lattice. We find a strong difference between Ramsey numbers on the Boolean lattice and ordered Ramsey numbers when the partial ordering on the graphs have large antichains.Comment: 18 pages, 3 figures, 1 tabl

    Bounded colorings of multipartite graphs and hypergraphs

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    Let cc be an edge-coloring of the complete nn-vertex graph KnK_n. The problem of finding properly colored and rainbow Hamilton cycles in cc was initiated in 1976 by Bollob\'as and Erd\H os and has been extensively studied since then. Recently it was extended to the hypergraph setting by Dudek, Frieze and Ruci\'nski. We generalize these results, giving sufficient local (resp. global) restrictions on the colorings which guarantee a properly colored (resp. rainbow) copy of a given hypergraph GG. We also study multipartite analogues of these questions. We give (up to a constant factor) optimal sufficient conditions for a coloring cc of the complete balanced mm-partite graph to contain a properly colored or rainbow copy of a given graph GG with maximum degree Δ\Delta. Our bounds exhibit a surprising transition in the rate of growth, showing that the problem is fundamentally different in the regimes Δm\Delta \gg m and Δm\Delta \ll m Our main tool is the framework of Lu and Sz\'ekely for the space of random bijections, which we extend to product spaces
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