The number of sparsely edged labelled Hamiltonian graphs

Abstract

An (n, q) graph is a graph on n labelled points and q lines, no loops and no multiple lines. We write N = ½n(n – 1), B(a, b) = a!/{b!(a – b)!} and B(a, 0) = 1, so that there are just B(N, q)different (n, q) graphs. Again h(n, q) is the number of Hamiltonian (n, q) graphs. Much attention has been devoted to the problem of determining for which q = q(n) “almost all” (n, q) graphs are Hamiltonian, i.e. for which q we haveas n → ∞. I proved [8, Theorem 4] that qn–3/2; → ∞ is a sufficient condition by showing that, for such q, almost all (n, q) graphs have about the average number of Hamiltonian circuits (H.c.s).