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Richard Mycroft (Birmingham, UK): Packing k-partite k-graphs |
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| Abstract. Let G and H be graphs or hypergraphs. A perfect H-packing in G is a collection of vertex-disjoint copies of H in G which together cover every vertex of G. In the simplest case, where H is the graph consisting of a single edge, a perfect H-packing in G is simply a perfect matching in G; Dirac's theorem tells us that such a packing must exist if G has minimum degree at least n/2 (where n is the number of vertices of G). The problem of what minimum degree is needed to ensure a perfect H-packing in G for general graphs H was then tackled by many researchers, before Kühn and Osthus finally established the correct threshold for all graphs H (up to an additive constant). However, for k-uniform hypergraphs (or k-graphs) much less is known. The case of a perfect matching has been well-studied, but apart from this there were previously no known asymptotically correct results on the minimum degree needed to ensure a perfect H-packing in G for k > 4 (for any of the various common generalisations of the notion of degree to the k-graph setting). In this talk I will demonstrate, for any complete k-partite k-graph H, the asymptotically best-possible minimum codegree condition for a k-graph G which ensures that G contains a perfect H-packing. This condition depends on the sizes of the vertex classes of H, and whether these sizes, or their differences, share any common factors greater than one. |
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| Location : Kalmár Intézet, Árpád tér, szemináriumi szoba | ||||
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