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Bipanconnectivity and edge-fault-tolerant bipancyclicity of hypercubes. (English) Zbl 1161.68684
Summary: A bipartite graph is bipancyclic if it contains a cycle of every even length from 4 to |V(G)| inclusive. It has been shown that Q n is bipancyclic if and only if n2. In this paper, we improve this result by showing that every edge of Q n -E ' lies on a cycle of every even length from 4 to |V(G)| inclusive where E ' is a subset of E(Q n ) with |E ' |n-2. The result is proved to be optimal. To get this result, we also prove that there exists a path of length l joining any two different vertices x and y of Q n when h(x,y)l|V(G)|-1 and l-h(x,y) is even where h(x,y) is the Hamming distance between x and y.
68R10Graph theory in connection with computer science (including graph drawing)