Hamiltonian cycles above expectation in r-graphs and quasi-random r-graphs

Abstract

Let $H_r(n,p)$ denote the maximum number of Hamiltonian cycles in an $n$-vertex $r$-graph with density $p\in (0,1)$. The expected number of Hamiltonian cycles in the random $r$-graph model $G_r(n,p)$ is $E(n,p)=p^n(n-1)!/2$ and in the random graph model $G_r(n,m)$ with $m=p\left(\genfrac{}{}{0ex}{}{n}{r}\right)$ it is, in fact, slightly smaller than $E(n,p)$. For graphs, $H_2(n,p)$ is proved to be only larger than $E(n,p)$ by a polynomial factor and it is an open problem whether a quasi-random graph with density $p$ can be larger than $E(n,p)$ by a polynomial factor. For hypergraphs (i.e. $r\ge 3$) the situation is drastically different. For all $r\ge 3$ it is proved that $H_r(n,p)$ is larger than $E(n,p)$ by an {\em exponential} factor and, moreover, there are quasi-random $r$-graphs with density $p$ whose number of Hamiltonian cycles is larger than $E(n,p)$ by an exponential factor.