A characterization of the edge connectivity of direct products of graphs

The direct product of graphs G=(V(G),E(G)) and H=(V(H),E(H)) is the graph, denoted as G×H, with vertex set V(G×H)=V(G)×V(H), where vertices (x1,y1) and (x2,y2) are adjacent in G×H if x1x2∈E(G) and y1y2∈E(H). The edge connectivity of a graph G, denoted as λ(G), is the size of a minimum edge-cut in G. We introduce a function ψ and prove the following formula λ(G×H)=min{2λ(G)|E(H)|,2λ(H)|E(G)|,δ(G×H),ψ(G,H),ψ(H,G)}. We also describe the structure of every minimum edge-cut in G×H.