Matching preclusion for direct product of regular graphs

Abstract

Let G be a graph with an even number of vertices. The matching preclusion number of G, denoted by mp(G), is the minimum number of edges whose deletion leaves the resulting graph without a perfect matching. G is maximally matched if mp(G) is equal to the minimum degree of G and is super matched if every optimal matching preclusion set of G consists of edges incident to a single vertex. In this paper, we focus on matching preclusion for direct product of graphs. For any two graphs G and H, we denote their direct product by G×H and show mp(G×H)⩾mp(G)mp(H), which implies that the direct product of two maximally matched graphs is also maximally matched. Furthermore, for any two regular graphs with at least three vertices, we show that if at least one of them is maximally matched, then their direct product is super matched, and as a consequence, their strong product is also super matched.