Fault-tolerant-prescribed hamiltonian laceability of balanced hypercubes
Section snippets
Introduction and preliminaries
For the graph-theoretical terminology, notation and operation used without defining here we follow [1] and [9]. Set for a positive integer n.
Let G be a bipartite graph with bipartition , and let u and v be two arbitrary vertices of X and Y, respectively. We refer to a path of G between u and v as , and refer to a “hamiltonian path” as an “H-path”. G is hamiltonian laceable if G has an H-path [6]. Let be an arbitrary set of at most f faulty edges. G is
Proof of Theorem 1.6
In this section, all additions and subtractions on some digit of a vertex in are modulo 4 unless otherwise stated.
Let be a set of faulty edges and let L be linear forest in such that . Let and such that and L are compatible. In the remainder of the section, we will prove that admits a fault free H-path passing through L. It suffices to show that has an H-path for the case that reaches its upper bound .
By
Concluding remarks
In this paper, we proved that the n-dimensional balanced hypercube is -fault-tolerant-prescribed hamiltonian laceable for , which extended Chen et al.'s result [2] and Lü et al.'s result [5]. Note that is a rectangle on the four vertices labelled by 0, 1, 2 and 3. Clearly, has no H-path between the vertices 1 and 2, which implies that the upper bound on cannot be improved for . The problem of the fault-tolerant-prescribed hamiltonian laceability of
Acknowledgement
The authors would like to express their heartfelt thanks to the referees for their comments and suggestions which are helpful to improve the quality of this paper. This work is partly supported by the Joint Fund of NSFC and Henan Province (U1304601).
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