Edge-bipancyclicity of a hypercube with faulty vertices and edges

A bipartite graph G=(V,E) is said to be bipancyclic if it contains a cycle of every even length from 4 to |V|. Furthermore, a bipancyclic G is said to be edge-bipancyclic if every edge of G lies on a cycle of every even length. Let F"v (respectively, F"e) be the set of faulty vertices (respectively, faulty edges) in an n-dimensional hypercube Q"n. In this paper, we show that every edge of Q"n-F"v-F"e lies on a cycle of every even length from 4 to 2^n-2|F"v| even if |F"v|+|F"e|==3. Since Q"n is bipartite of equal-size partite sets and is regular of vertex-degree n, both the number of faults tolerated and the length of a longest fault-free cycle obtained are worst-case optimal.

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