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

Let r(2) >= r(1) >= 0 be two integers. A bipartite graph G is two-disjoint-cycle-cover vertex [r(1), r(2)]-bipancyclic (2-DCC vertex [r(1), r(2)]-bipancyclic in short) if for any two vertices u, v is an element of V (G) and any even integer l satisfying r(1) <= l <= r(2), there exist two vertex-disjoint cycles J(1) and J(2) in G with vertical bar V(J(1))vertical bar = l and vertical bar V(J(2))vertical bar = vertical bar V(G)vertical bar - l such that u is an element of V(J(1)) and v is an element of V(J(2)); and there also exist two vertex-disjoint cycles J(1)' and J(2)' in G with vertical bar V(J(1)')vertical bar = l and vertical bar V(J(2)')vertical bar = vertical bar V(G)vertical bar - l such that v is an element of V(J(1)') and u is an element of V(J(2)'). We study the 2-DCC vertex bipancyclicity of the n-dimensional bipartite generalized hypercube C(d(1), d(2), ..., d(0)). As a result, we deter- mine 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, vertical bar V(G)vertical bar/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. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

The text discusses the study of specific bipancyclic properties in bipartite graphs, focusing on generalized hypercubes and k-ary n-cubes. It examines the 2-DCC vertex bipancyclicity and identifies exceptional graphs, showing that certain criteria must be met for a n-dimensional bipartite generalized hypercube to be considered 2-DCC vertex [4, |V(G)|/2]-bipancyclic. Additionally, it demonstrates the vertex-bipancyclicity and 2-DCC bipancyclicity properties in n-dimensional bipartite generalized hypercubes and extends these properties to n-dimensional bipartite k-ary n-cubes.