The number of cycles in a hamilton graph

Abstract The set of Hamilton graphs (having no loops) with n(⩾2) vertices and n+k edges is denoted by Γk and the number of distinct cycles of a graph G is denoted by ƒG) . Let m(k)= min {ƒ(G); G∈Γ k } and M(k)=max {z.hfl;(G); Gisin;Γk}. 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)⩽m⩽M(k), can we find a graph G from Гk such that ƒ(G)=m? In this paper we answer the first question in the affirmative and the others in the negative.