Vertex-bipancyclicity Of The Generalized Honeycomb Tori

摘要

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 graph C 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 from 4to |V(G)| that passes v. A bipartite graph C 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.