Connectivity of Kronecker products by

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 ) , v 1 v 2 ∈ E ( G 2 ) } . In this paper, we prove that κ ( G × K 2 ) = min { 2 κ ( G ) , min { | X | + 2 | Y | } } , where the second minimum is taken over all disjoint sets X , Y ⊆ V ( G ) satisfying (1) G − ( X ∪ Y ) has a bipartite component C , and (2) G [ V ( C ) ∪ { x } ] is also bipartite for any x ∈ X .