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 ( x 1 , y 1 ) and ( x 2 , y 2 ) are adjacent in G × H if x 1 x 2 ∈ E ( G ) and y 1 y 2 ∈ 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 { 2 λ ( G ) | E ( H ) | , 2 λ ( 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 .