Pancyclism and Bipancyclism of Hamiltonian Graphs

In this paper, we prove the following two theorems: (1) If G is a hamiltonian graph of order n and if there exists a vertex x ∈ V(G) such that d(x) + d(y) ≥ n for each y not adjacent to x, then G is either pancyclic or the complete bipartite graph K(n/2, n/2). (2) Let G=(X, Y; E) be a hamiltonian bipartite graph with |X| = |Y| = n > 3. If there exists a vertex x ∈ X such that d(x) + d(y) ≥ n + 1 for each y ∈ Y not adjacent to x, then G is bipancyclic. The bounds in the two theorems are best possible.