A note on the connectivity of Kronecker products of graphs

Let κ ( G ) be the connectivity of G . The Kronecker product G 1 × G 2 of graphs G 1 and G 2 has vertex set V ( G 1 × G 2 ) = V ( G 1 ) × V ( G 2 ) and edge set E ( G 1 × G 2 ) = { ( u 1 , v 1 ) ( u 2 , v 2 ) : u 1 u 2 ∈ E ( G 1 )  and  v 1 v 2 ∈ E ( G 2 ) } . In this paper, we prove that κ ( G × K n ) = min { n κ ( G ) , ( n − 1 ) δ ( G ) } , where G is a bipartite graph.