17,315 research outputs found

    Embedding cycles of given length in oriented graphs

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    Kelly, Kuehn and Osthus conjectured that for any l>3 and the smallest number k>2 that does not divide l, any large enough oriented graph G with minimum indegree and minimum outdegree at least \lfloor |V(G)|/k\rfloor +1 contains a directed cycle of length l. We prove this conjecture asymptotically for the case when l is large enough compared to k and k>6. The case when k<7 was already settled asymptotically by Kelly, Kuehn and Osthus.Comment: 8 pages, 2 figure

    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

    Resolution of the Oberwolfach problem

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    The Oberwolfach problem, posed by Ringel in 1967, asks for a decomposition of K2n+1K_{2n+1} into edge-disjoint copies of a given 22-factor. We show that this can be achieved for all large nn. We actually prove a significantly more general result, which allows for decompositions into more general types of factors. In particular, this also resolves the Hamilton-Waterloo problem for large nn.Comment: 28 page

    Acyclic Subgraphs of Planar Digraphs

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    An acyclic set in a digraph is a set of vertices that induces an acyclic subgraph. In 2011, Harutyunyan conjectured that every planar digraph on nn vertices without directed 2-cycles possesses an acyclic set of size at least 3n/53n/5. We prove this conjecture for digraphs where every directed cycle has length at least 8. More generally, if gg is the length of the shortest directed cycle, we show that there exists an acyclic set of size at least (1−3/g)n(1 - 3/g)n.Comment: 9 page
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