The generalized 3-connectivity of Lexicographic product graphs

The generalized $k$-connectivity $κ_k(G)$ of a graph $G$, introduced by Chartrand et al., is a natural and nice generalization of the concept of (vertex-)connectivity. In this paper, we prove that for any two connected graphs $G$ and $H$, $κ_3(G\circ H)\geq κ_3(G)|V(H)|$. We also give upper bounds for $κ_3(G\Box H)$ and $κ_3(G\circ H)$. Moreover, all the bounds are sharp.

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