The Matching Preclusion of Enhanced Hypercubes

Abstract

Abstract The (conditional) matching preclusion number of a graph is the minimum number of edges whose deletion leaves the resulting graph (with no isolated vertices) that has neither perfect matchings nor almost perfect matchings. The (conditional) strong matching preclusion number of a graph is the minimum number of vertices and edges whose deletion makes the resulting graph (with no isolated vertices) without perfect matching or almost perfect matching. The enhanced hypercube $Q_{n,k}$ $(1\leq k\leq n-1)$ is an extension of hypercube. In this paper, we prove that the matching preclusion number of $Q_{n,k}$ is $n+1$ $(1\leq k\leq n-1)$, the strong matching preclusion number of $Q_{n,k}$ is $n+1$ $(2\leq k\leq n-1)$, the conditional matching preclusion number of $Q_{n,n-1}$ is $2n-1$, the conditional matching preclusion number of $Q_{n,k}$ is $2n$ $(1\leq k\leq n-2)$ and the conditional strong matching preclusion number of $Q_{n,n-2}$ is $2n-3$ $(n\geq 4)$.