Vertex-bipancyclicity of the generalized honeycomb tori

Assume that m, n and s are integers with m ≥ 2, n ≥ 4, 0 ≤ s < n and s is of the same parity of m. The generalized honeycomb tori GHT(m, n, s) have been recognized as an attractive architecture to existing torus interconnection networks in parallel and distributed applications. A bipartite graphG is bipancyclic if it contains a cycle of every even length from 4 to |V (G)| inclusive. G is vertex-bipancyclic if for any vertex v ∈ V (G), there exists a cycle of every even length from4 to |V (G)| that passes v. A bipartite graphG is called k-vertex-bipancyclic if every vertex lies on a cycle of every even length from k to |V (G)|. In this article, we prove that GHT(m, n, s) is 6-bipancyclic, and is bipancyclic for some special cases. Since GHT(m, n, s) is vertex-transitive, the result implies that any vertex of GHT(m, n, s) lies on a cycle of length l, where l ≥ 6 and is even. Besides, GHT(m, n, s) is vertex-bipancyclic in some special cases. The result is optimal in the sense that the absence of cycles of certain lengths on some GHT(m, n, s)’s is inevitable due to their hexagonal structure. © 2008 Elsevier Ltd. All rights reserved.