# Two bounds for generalized $3$-connectivity of Cartesian product graphs

## TL;DR

This paper establishes two new lower bounds for the generalized 3-connectivity of Cartesian product graphs, one surpassing previous bounds and the other confirming a conjecture for 3-connected graphs.

## Contribution

It provides two distinct lower bounds for the generalized 3-connectivity of Cartesian product graphs, improving upon existing results and validating a conjecture.

## Key findings

- First lower bound is stronger than previous bounds.
- Second lower bound confirms the conjecture for 3-connected graphs.
- Advances understanding of connectivity in product graphs.

## Abstract

The generalized $k$-connectivity $\kappa_{k}(G)$ of a graph $G$, which was introduced by Chartrand et al.(1984) is a generalization of the concept of vertex connectivity. Let $G$ and $H$ be nontrivial connected graphs. Recently, Li et al. gave a lower bound for the generalized $3$-connectivity of the Cartesian product graph $G \square H$ and proposed a conjecture for the case that $H$ is $3$-connected. In this paper, we give two different forms of lower bounds for the generalized $3$-connectivity of Cartesian product graphs. The first lower bound is stronger than theirs, and the second confirms their conjecture.