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 ∈ (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 n r 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 ≥ 3) the situation is drastically different. For all r ≥ 3 it is proved that H r (n, p) is larger than E(n, p) by an exponential factor and, moreover, there are quasirandom r-graphs with density p whose number of Hamiltonian cycles is larger than E(n, p) by an exponential factor.