Hamiltonian paths and cycles pass through prescribed edges in the balanced hypercubes

Abstract

The n-dimensional balanced hypercube BHn (n≥1) has been proved to be a bipartite graph. Let P be a set of edges in BHn. For any two vertices u,v from different partite sets of V(BHn). In this paper, we prove that if |P|≤2n−2 and the subgraph induced by P has neither u nor v as internal vertices , or both of u and v as end-vertices, then BHn contains a Hamiltonian path joining u and v passing through every edge of P if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths. As a corollary, if |P|≤2n−1, then BHn contains a Hamiltonian cycle passing through every edge of P if and only if the subgraph induced by P consists of pairwise vertex-disjoint paths.