Connectivity and Super-Connectivity of Cartesian Product Graphs

Abstract

This paper determines that the connectivity of the Cartesian product \(G_1 \square G_2\) of two graphs \(G_1\) and \(G_2\) is equal to \(\min\{\kappa_1v_2 + \kappa_2v_1, \delta_1 + \delta_2 \}\), where \(v_i, \kappa_i\), and \(\delta_i\) are the order, connectivity, and minimum degree of \(G_i\), respectively, for \(i = 1, 2\). Additionally, some necessary and sufficient conditions are given for \(G_1 \square G_2\) to be maximally connected and super-connected.