Two-disjoint-cycle-cover vertex bipancyclicity of bipartite hypercube-like networks

Abstract

Let r1,r2 be two integers such that r2≥r1≥0. A bipartite graph G is two-disjoint-cycle-cover vertex [r1,r2]-bipancyclic (2-DCC vertex [r1,r2]-bipancyclic for short) if for any two vertices u,v∈V(G) and any even integer ℓ satisfying r1≤ℓ≤r2, there exist two vertex-disjoint cycles C1 and C2 in G with |V(C1)|=ℓ and |V(C2)|=|V(G)|−ℓ such that u∈V(C1) and v∈V(C2). In this paper, we study the 2-DCC vertex bipancyclicity of the n-dimensional bipartite hypercube-like network, which is one class of hypercube-generalized networks. As a consequence, we show that an n-dimensional bipartite hypercube-like network is 2-DCC vertex [4,2n−1]-bipancyclic for n≥3. In particular, it provides an application that n-dimensional hypercube and bicube are also 2-DCC vertex [4,2n−1]-bipancyclic for n≥3.