Two-disjoint-cycle-cover vertex bipancyclicity of the bipartite generalized hypercube

Let r2≥r1≥0 be two integers. A bipartite graph G is two-disjoint-cycle-cover vertex [r1,r2]-bipancyclic (2-DCC vertex [r1,r2]-bipancyclic in 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 J1 and J2 in G with |V(J1)|=ℓ and |V(J2)|=|V(G)|−ℓ such that u∈V(J1) and v∈V(J2); and there also exist two vertex-disjoint cycles J1′ and J2′ in G with |V(J1′)|=ℓ and |V(J2′)|=|V(G)|−ℓ such that v∈V(J1′) and u∈V(J2′). We study the 2-DCC vertex bipancyclicity of the n-dimensional bipartite generalized hypercube C(d1,d2,…,dn). As a result, we determine a family of exceptional graphs and show that for all integers n≥2, an n-dimensional bipartite generalized hypercube G is 2-DCC vertex [4,|V(G)|/2]-bipancyclic if and only if G is not a member in this family. Furthermore, as applications, we prove the vertex-bipancyclicity and 2-DCC bipancyclicity on n-dimensional bipartite generalized hypercube and show that the similar properties also hold for all n-dimensional bipartite k-ary n-cubes, for n≥2.