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arXiv · 0903.2867

Hamilton l-cycles in uniform hypergraphs

Daniela Kühn·Richard Mycroft·Deryk Osthus
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Abstract

We say that a k-uniform hypergraph C is an l-cycle if there exists a cyclic ordering of the vertices of C such that every edge of C consists of k consecutive vertices and such that every pair of consecutive edges (in the natural ordering of the edges) intersects in precisely l vertices. We prove that if 1 \leq l \leq k-1 and k-l does not divide k then any k-uniform hypergraph on n vertices with minimum degree at least n/((\lceil (k/(k-l)) \rceil)(k-l))+o(n) contains a Hamilton l-cycle. This confirms a conjecture of H\`an and Schacht. Together with results of R\"odl, Ruci\'nski and Szemer\'edi, our result asymptotically determines the minimum degree which forces an l-cycle for any l with 1 \leq l \leq k-1.

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Daniela Kühn, Richard Mycroft, Deryk Osthus. 2009-03-16. Hamilton l-cycles in uniform hypergraphs. https://doi.org/10.1016/j.jcta.2010.02.010

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