Cycles embedding in balanced hypercubes with faulty edges and vertices

摘要

Wu and Huang proposed a new variation of the hypercube, named balanced hypercube, which possesses many good properties such as bipanconnectivity, edge-bipancyclicity, Hamiltonian laceability, hyper Hamiltonian laceability. In this paper, we consider n-dimensional balanced hypercube with vertical bar F-e vertical bar faulty edges and vertical bar F-v vertical bar faulty vertices. We prove that if vertical bar F-v vertical bar + vertical bar F-e vertical bar = n - 1, then every fault-free edge of BHn lies on a fault-free cycle of every even length from 6 to 2(2n) -2 vertical bar F-v vertical bar, where n = 2; and if vertical bar F-v vertical bar + vertical bar F-e vertical bar = 2n - 3, then there is a fault-free cycle of every even length from 6 to 2(2n) -2 vertical bar F-v vertical bar in BHn, where n = 2. Furthermore, we propose the distance between vertex-disjoint edge e and cycle C, i.e., d(e, C) = min{d(e, e') vertical bar e' is an element of E(C)}, where d(e, e') = min{d(u, x), d(u, y), d(v, x), d(v, y) I (u, v) = e, (x, y) = e'}. (C) 2017 Elsevier B.V. All rights reserved.