Hamiltonian paths passing through prescribed edges in balanced hypercubes

The balanced hypercube was proposed as a novel interconnection network for large-scale parallel systems. It is known that the balanced hypercube is edge-bipancyclic. Given a set of edges P with vertical bar P vertical bar <= n - 1 and two vertices x and y in different bipartite sets, we show that there exists a Hamiltonian path from x to y passing through P in BHn, if and only if P is a linear forest and, neither x nor y is an internal vertex of P, and x and y are not end-vertices of a component of P simultaneously. Consequently, the balanced hypercube contains a Hamiltonian cycle passing through a set P of at most n prescribed edges if and only if P is a linear forest. Moreover, our result improves some known results. (C) 2018 Elsevier B.V. All rights reserved.