Conditional matching preclusion for regular bipartite graphs and their Cartesian product

Abstract

Let G be a graph with an even number of vertices. The conditional matching preclusion number of G is the minimum number of edges whose deletion results in a graph with neither isolated vertices nor perfect matchings. Conditional matching preclusion number was introduced as an improvement of matching preclusion number for measuring the robustness of a network when there is a link failure with a higher accuracy. In this paper, we characterize all conditionally maximally matched regular bipartite graphs and conditionally super matched regular bipartite graphs, respectively. Furthermore, for r1-regular bipartite graph G and r2-regular bipartite graph H with r1⩾3 and r2⩾3, we show that if G is conditionally maximally matched and H is r2∕2-edge-connected, then G□H is conditionally super matched except for one class of graphs.