Counting Hamilton Cycles in Dirac Hypergraphs

Abstract

For 0le ell <k, a Hamilton ell -cycle in a k-uniform hypergraph H is a cyclic ordering of the vertices of H in which the edges are segments of length k and every two consecutive edges overlap in exactly ell vertices. We show that for all 0le ell <k-1, every k-graph with minimum co-degree delta n with delta >1/2 has (asymptotically and up to a subexponential factor) at least as many Hamilton ell -cycles as a typical random k-graph with edge-probability delta . This significantly improves a recent result of Glock, Gould, Joos, Kühn and Osthus, and verifies a conjecture of Ferber, Krivelevich and Sudakov for all values 0le ell <k-1.