Edge-bipancyclicity of hypercubes with faulty edges

Abstract

In this paper, we consider the edge-bipancyclicity property of hypercubes with faulty edges. For $n\ge 4$, let $Q_n$ be an n-dimensional hypercube, and let F be a set of faulty edges in $Q_n$ such that $\left|F\right|\le 3n-11$. If every vertex in $Q_n-F$ has a degree of at least 2 and no 4-cycle in $Q_n-F$ has a pair of non-adjacent vertices both of degree 2, then every edge e in $Q_n-F$ lies on a fault-free cycle of every even length from 8 to $2^n$, with two exceptions: (1) there exists a 4-cycle in $Q_n-F$ containing the edge e such that the vertices on the 4-cycle which are not incident with e both have a degree of 2; (2) there exist two vertices of degree 2 which are not endpoints of edge e but are adjacent to the same endpoint of e in $Q_n-F$. This result improves upon some known results on edge-bipancyclicity of hypercubes with faulty edges.