The edge-pancyclicity of dual-cube extensive networks

The hypercube family Q(n) is one of the most well-known interconnection networks in parallel computers, e.g. see [3, 6]. Using Q(n) Li et al. introduced dual-cube networks DC(n) and show the vertex symmetry and some fault-tolerant hamiltonian properties of DC(n) [7, 8]. By replacing Q with any arbitrary graph G in DC(n), a new family of interconnection networks called dual-cube extensive networks, denoted by DCEN(G), was introduced recently [4]. It is proved in [4] that DCEN(G) preserves the hamiltonian connectivity and the globally-3*-connectivity of G. In this article, we prove that DCEN(G) "preserves" the edge-pancyclicity of G as well. More precisely, suppose that every edge G lies on a cycle of length 1, where 1 is an arbitrary integer with 3 <= 1 <= vertical bar G vertical bar. Then for any edge (e) over cap of DCEN(G), there exists a cycle in DCEN(G), denoted by (C) over cap (l) such that (e) over cap is an element of (C) over cap (l) and the length of (C) over cap (l) is l for every l(min) <= l <= vertical bar DCEN(G)vertical bar, where l(min) is an element of {3,8}. The result is shown to be optimal. Furthermore, we prove that the similar results hold when G is a bipartite graph.