Bipancyclic Properties of Faulty Hypercubes

A bipartite graph G = ( V , E ) is bipancyclic if it contains cycles of every even length from 4 to | V | and edge bipancyclic if every edge lies on a cycle of every even length from 4 to | V | . LetQ ndenote the n -dimensional hypercube. Let F be a subset of V ( Q n ) ∪ E ( Q n ) such that F can be decomposed into two partsF a vandF e, whereF a vis a union off a vdisjoint adjacent pairs of V ( Q n ) , andF econsists off eedges. We prove thatQ n - F is bipancyclic iff a v + f e ≤ n - 2 . Moreover,Q n - F is edge bipancyclic iff a v + f e ≤ n - 2 withf a v < n - 2 .