Hamilton l-cycles in uniform hypergraphs

Details Hamilton l-cycles in uniform hypergraphs Hamilton l-cycles in uniform hypergraphs

2009-03-16

arxiv.org/abs/0903.2867v2

Mathematics

Combinatorics

v2: minor corrections. Accepted for publication in J. Combinatorial Theory, Series A. 20 pages

Scientific paper

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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