Matching preclusion for vertex-transitive networks

TL;DR

This paper studies the robustness of vertex-transitive networks against link failures by analyzing their matching preclusion numbers, revealing that most are super matched except for six classes, with new results for specific network types.

Contribution

It provides general results on matching preclusion for vertex-transitive graphs, identifying exceptions and deriving new results for specific network classes.

Findings

Matching preclusion number equals degree for most vertex-transitive graphs.

Most vertex-transitive graphs are super matched, except for six classes.

New matching preclusion results for folded k-cubes, Hamming graphs, and halved k-cubes.

Abstract

In interconnection networks, matching preclusion is a measure of robustness when there is a link failure. Let $G$ be a graph of even order. The matching preclusion number $mp(G)$ is defined as the minimum number of edges whose deletion results in a subgraph without perfect matchings. Many interconnection networks are super matched, that is, their optimal matching preclusion sets are precisely those induced by a single vertex. In this paper, we obtain general results of vertex-transitive graphs including many known networks. A $k$-regular connected vertex-transitive graph has matching preclusion number $k$ and is super matched except for six classes of graphs. From this many previous results can be directly obtained and matching preclusion for some other networks, such as folded $k$-cubes, Hamming graphs and halved $k$-cubes, are derived.