Integer k-matching preclusion of some interconnection networks

Abstract

For a positive integer k, a k-matching of a graph G is a function f: E(G)→{0,1,…,k} such that ∑e∈Γ(v)f(e)≤k for every vertex v of G, where Γ(v) represents the set of all edges incident to v. The (strong) k-matching preclusion number of G, denoted by (smpk(G))mpk(G), is the minimum size of (edges and vertices) edges whose deletion leaves the remaining subgraph that has neither perfect k-matchings nor almost perfect k-matchings. This is a generalization of the concept matching preclusion problem proposed by Brigham et al. In this paper, we consider the k-matching preclusion problem of some interconnection networks, such as the hypercube-like graphs HLn, the augmented cubes AQn and the torus networks Cm1□Cm2□⋯□Cmn, which are variants of the hypercube graphs Qn. Our result on restricted hypercube-like graphs RHLn generalizes the result of twisted cubes TQn by Chang et al. [3].