Counting Hamilton Cycles in Dirac Hypergraphs

Abstract

AbstractFor $$0\le \ell <k$$ 0 ≤ ℓ < k , a Hamilton $$\ell $$ ℓ -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 $$\ell $$ ℓ vertices. We show that for all $$0\le \ell <k-1$$ 0 ≤ ℓ < k - 1 , every k-graph with minimum co-degree $$\delta n$$ δ n with $$\delta >1/2$$ δ > 1 / 2 has (asymptotically and up to a subexponential factor) at least as many Hamilton $$\ell $$ ℓ -cycles as a typical random k-graph with edge-probability $$\delta $$ δ . 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 $$0\le \ell <k-1$$ 0 ≤ ℓ < k - 1 .