The Super Restricted Edge-connectedness of Direct Product Graphs

Abstract

Let [Formula: see text] be a graph with vertex set [Formula: see text] and edge set [Formula: see text]. An edge subset [Formula: see text] is called a restricted edge-cut if [Formula: see text] is disconnected and has no isolated vertices. The restricted edge-connectivity [Formula: see text] of [Formula: see text] is the cardinality of a minimum restricted edge-cut of [Formula: see text] if it has any; otherwise [Formula: see text]. If [Formula: see text] is not a star and its order is at least four, then [Formula: see text], where [Formula: see text]. The graph [Formula: see text] is said to be maximally restricted edge-connected if [Formula: see text]; the graph [Formula: see text] is said to be super restricted edge-connected if every minimum restricted edge-cut isolates an edge from [Formula: see text]. The direct product of graphs [Formula: see text] and [Formula: see text], denoted by [Formula: see text], is the graph with vertex set [Formula: see text], where two vertices [Formula: see text] and [Formula: see text] are adjacent in [Formula: see text] if and only if [Formula: see text] and [Formula: see text]. In this paper, we give a sufficient condition for [Formula: see text] to be super restricted edge-connected, where [Formula: see text] is the complete graph on [Formula: see text] vertices.