On the Path-Connectivity of Lexicographic Product Graphs

Abstract

The \(k\)-path-connectivity \(\pi_k(G)\) of a graph \(G\) was introduced by Hager in \(1986\). Recently, Mao investigated the \(3\)-path-connectivity of lexicographic product graphs. Denote by \(G \circ H\) the lexicographic product of two graphs \(G\) and \(H\). In this paper, we prove that \(\pi_4(G \circ H) \geq \lfloor\frac{|V(H)|-2}{3}\rfloor\) for any two connected graphs \(G\) and \(H\). Moreover, the bound is sharp. We also derive an upper bound of \(\pi_4(G \circ H)\), that is, \(\pi_4(G \circ H) \leq 2\pi_4(G)|V(H)|\).