Edge-bipancyclicity of conditional faulty hypercubes

Abstract

Xu et al. showed that for any set of faulty edges F of an n-dimensional hypercube Q n with | F | ⩽ n − 1 , each edge of Q n − F lies on a cycle of every even length from 6 to 2 n , n ⩾ 4 , provided not all edges in F are incident with the same vertex. In this paper, we find that under similar condition, the number of faulty edges can be much greater and the same result still holds. More precisely, we show that, for up to | F | = 2 n − 5 faulty edges, each edge of the faulty hypercube Q n − F lies on a cycle of every even length from 6 to 2 n with each vertex having at least two healthy edges adjacent to it, for n ⩾ 3 . Moreover, this result is optimal in the sense that there is a set F of 2 n − 4 conditional faulty edges in Q n such that Q n − F contains no hamiltonian cycle.