A CONDITION FOR A HAMILTONIAN BIPARTITE GRAPH TO BE BIPANCYCLIC

Let G be a hamiltonian bipartite graph of order 2n and let C = (x1, y1, x2, y2, . . . , x(n), y(n), x1) be a hamiltonian cycle of G. G is said to be bipancyclic if it contains a cycle of length 2l, for every 1, 2 less-than-or-equal-to l less-than-or-equal-to n. Suppose the vertices x, and X2 are such that d(x1) + d(X2) greater-than-or-equal-to n + 1. Then G is either: (1) bipancyclic, (2) missing a 4-cycle (then n is odd and the structure of G is known), (3) missing a (n + 1)-cycle (then n is odd and the structure of G is known).