On path bipancyclicity of hypercubes

Assume that P is any path in a bipartite graph G of length k with 2 <= k <= h, G is said to be h-path bipancyclic if there exists a cycle C in G of every even length from 2k to |V(G)| such that P lies in C. Based on Lemma 5, the authors of [C.-H. Tsai, S.-Y.Jiang, Path bipancyclicity of hypercubes, Inform. Process. Lett. 101 (2007) 93-97] showed that the n-cube Q(n) with n >= 3 is (2n - 4)-path bipancyclicity. In this paper, counterexamples to the lemma are given, therefore, their proof fails. And we show the following result: The n-cube Q(n) with n >= 3 is (2n - 4)-path bipancyclicity but is not (2n - 2)-path bipancyclicity, moreover, and a path P of length k with 2 <= k <= 2n - 4 lies in a cycle of length 2k - 2 if and only if P contains two edges of dimension i for some i, 1 <= i <= n. We conjecture that if 2n - 4 is replaced by 2n - 3, then the above result also holds. (c) 2009 Elsevier B.V. All rights reserved.