The Super Restricted Edge- connectedness of Direct Product Graphs

Let G be a graph with vertex set V (G) and edge set E(G). An edge subset F subset of E(G) is called a restricted edge-cut if G - F is disconnected and has no isolated vertices. The restricted edge-connectivity lambda' (G) of G is the cardinality of a minimum restricted edgecut of G if it has any; otherwise lambda'(G) = +infinity. If G is not a star and its order is at least four, then lambda' (G) <= (G), where xi(G) = min{d (G) (u)+d (G) (v) - 2 : uv is an element of E(G)}. The graph G is said to be maximally restricted edge-connected if lambda' (G) = xi(G); the graph G is said to be super restricted edge-connected if every minimum restricted edge-cut isolates an edge from G. The direct product of graphs G (1) and G (2), denoted by G (1) x G (2), is the graph with vertex set V (G (1) x G (2)) = V (G (1)) x V (G (2)), where two vertices (u1, v 1) and (u2, v 2) are adjacent in G (1) x G (2) if and only if u (1)u (2) is an element of E(G (1)) and v (1) v (2) is an element of E(G (2)). In this paper, we give a sufficient condition for G x K (n) to be super restricted edge-connected, where K n is the complete graph on n vertices.