A note on the connectivity of the Cartesian product of graphs

We present an example of two connected graphs for which the connectivity of the Cartesian product of the graphs is strictly greater than the sum of the connectivities of the factor graphs. This clarifies an issue from the literature. We then find necessary and sufficient conditions for the connectivity of the Cartesian product of the graphs to be equal to the sum of the connectivities of the individual graphs. Throughout this paper we strive to use general terminology in graph theory from [3], and terminology concerning Cartesian products of graphs from [4]. Let G =(V (G),E(G)) be a graph, and let δ(G) denote the minimum degree among all vertices in G. The connectivity κ(G)ofG is the minimum size of S ⊆ V (G) such that G − S is disconnected or a single vertex. The Cartesian product G✷H of two graphs G and H is the graph with vertex set V (G✷H) =V (G) × V (H), and edge set E(G✷H) containing all pairs of the form [(g1,h1), (g2,h2)] such that either [g1,g2] isanedgeinG and h1 = h2, or[h1,h2] i