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

Abstract

Let Hr(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 Gr(n,p) is E(n,p)=pn(n−1)!/2 and in the random graph model Gr(n,m) with m=p(nr) it is, in fact, slightly smaller than E(n,p).For graphs, H2(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 Hr(n,p) is larger than E(n,p) by an 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.