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

The n -dimensional balanced hypercube B H n (n ≥ 1) has been proved to be a bipartite graph. Let P be a set of edges in B H n. For any two vertices u , v from different partite sets of V (B H n). In this paper, we prove that if | P | ≤ 2 n − 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 B H n 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 | ≤ 2 n − 1 , then B H n 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. [ABSTRACT FROM AUTHOR]