Another cycle structure theorem for hamiltonian graphs

A graph G is pancyclic if it contains a k-cycle for k = 3,4,…,|V(G)|. In this paper we show a cycle theorem as follows: If C is a hamiltonian cycle of a graph G of order n, where two non-adjacent vertices x,y at distance 2 on C satisfy d(x)+d(y) ⩾ n, then G is either pancyclic, bipartite, missing only the (n − 1)-cycle, or missing the 3-cycle.