A condition for a hamiltonian bipartite graph to be bipancyclic

Abstract

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