Bipanconnectivity and edge-fault-tolerant bipancyclicity of hypercubes

Abstract

A bipartite graph is bipancyclic if it contains a cycle of every even length from 4 to I V(G) I inclusive. It has been shown that Q(n) is bipancyclic if and only if n greater than or equal to 2. In this paper, we improve this result by showing that every edge of Q(n) - E' lies on a cycle of every even length from 4 to V(G) inclusive where E' is a subset of E(Q(n)) with E' less than or equal to n - 2. The result is proved to be optimal. To get this result, we also prove that there exists a path of length I joining any two different vertices x and y of Qn when h (x, y) less than or equal to l less than or equal to V(G) - 1 and l - h (x, y) is even where It (x, y) is the Hamming distance between x and y. (C) 2003 Elsevier Science B.V. All rights reserved.