Proof of a conjecture on connectivity of Kronecker product of graphs

For a graph G , κ ( G ) denotes its connectivity. The Kronecker product G 1 × G 2 of graphs G 1 and G 2 is the graph with the vertex set V ( G 1 ) × V ( G 2 ) , two vertices ( u 1 , v 1 ) and ( u 2 , v 2 ) being adjacent in G 1 × G 2 if and only if u 1 u 2 ∈ E ( G 1 ) and v 1 v 2 ∈ E ( G 2 ) . Guji and Vumar [R. Guji, E. Vumar, A note on the connectivity of Kronecker products of graphs, Appl. Math. Lett. 22 (2009) 1360–1363] conjectured that for any nontrivial graph G , κ ( G × K n ) = min { n κ ( G ) , ( n − 1 ) δ ( G ) } when n ≥ 3 . In this note, we confirm this conjecture to be true.