Hamilton l-cycles in uniform hypergraphs

TL;DR

This paper proves a minimum degree condition that guarantees the existence of Hamilton l-cycles in k-uniform hypergraphs, confirming a conjecture and extending understanding of cycle thresholds in hypergraph theory.

Contribution

It establishes a new minimum degree threshold for Hamilton l-cycles in k-uniform hypergraphs, confirming a conjecture and generalizing previous results.

Findings

Proves minimum degree condition for Hamilton l-cycles when k-l does not divide k.

Confirms a conjecture of H ext`an and Schacht.

Asymptotically determines cycle thresholds for all l in 1 to k-1.

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.