Bipanconnectivity of faulty hypercubes with minimum degree

Abstract

In this paper, we consider the conditionally faulty hypercube Q n with n ⩾ 2 where each vertex of Q n is incident with at least m fault-free edges, 2 ⩽ m ⩽ n − 1. We shall generalize the limitation m ⩾ 2 in all previous results of edge-bipancyclicity. We also propose a new edge-fault-tolerant bipanconnectivity called k-edge-fault-tolerant bipanconnectivity. A bipartite graph is k-edge-fault-tolerant bipanconnected if G − F remains bipanconnected for any F ⊂ E( G) with ∣ F∣ ⩽ k. For every integer m, under the same hypothesis, we show that Q n is ( n − 2)-edge-fault-tolerant edge-bipancyclic and bipanconnected, and the results are optimal with respect to the number of edge faults tolerated. This not only improves some known results on edge-bipancyclicity and bipanconnectivity of hypercubes, but also simplifies the proof.