THE NUMBER OF CYCLES IN A HAMILTON GRAPH

The set of Hamilton graphs (having no loops) with n (greater-than-or-equal-to 2) vertices and n + k edges is denoted by GAMMA(k) and the number of distinct cycles of a graph G is denoted by f(G). Let m(k) = min {f(G); G is-an-element-of GAMMA(k)} and M(k) = max {f(G); G is-an-element-of GAMMA(k)}. Yap and Teo (1984) raised the following questions: (1) Is it true that m(k) = (k + 1)(k + 2)/2? (2) Is it true that M(k) = 2k + k? (3) For each integer m satisfying m(k) less-than-or-equal-to m less-than-or-equal-to M(k), can we find a graph G from GAMMA(k) such that f(G) = m? In this paper we answer the first question in the affirmative and the others in the negative.