Bipanconnectivity and edge-fault-tolerant bipancyclicity of hypercubes

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Abstract

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 Qn is bipancyclic if and only if n⩾2. In this paper, we improve this result by showing that every edge of QnE′ lies on a cycle of every even length from 4 to |V(G)| inclusive where E′ is a subset of E(Qn) 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 Qn when h(x,y)⩽l⩽|V(G)|−1 and lh(x,y) is even where h(x,y) is the Hamming distance between x and y.

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    When embedding paths and cycles into faulty hypercubes, the faulty nodes and/or edges should be avoided. Thus, the problem of embedding paths and cycles into faulty hypercubes has been proposed and investigated in depth (see for example, [2,4,5,8,9,12,16,17,20,21]). In particular, Sun et al. [20] studied the problem of embedding hamiltonian cycles and hamiltonian paths in hypercubes with faulty edges and/or disjoint adjacent node pairs.

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This work was supported in part by the National Science Council of the Republic of China under Contract NSC 91-2218-E-231-002.

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