# An improved bound in Vizing's conjecture

## TL;DR

This paper improves a lower bound related to Vizing's conjecture on domination numbers in Cartesian product graphs, strengthening previous results with a tighter inequality.

## Contribution

The authors present a new lower bound for the domination number of the Cartesian product of graphs, advancing the understanding of Vizing's conjecture.

## Key findings

- Established a new lower bound involving max of domination numbers
- Improved previous bounds by Suen and Tarr
- Contributed to the theoretical understanding of graph domination

## Abstract

A well-known conjecture of Vizing is that $\gamma(G \square H) \ge \gamma(G)\gamma(H)$ for any pair of graphs $G, H$, where $\gamma$ is the domination number and $G \square H$ is the Cartesian product of $G$ and $H$. Suen and Tarr, improving a result of Clark and Suen, showed $\gamma(G \square H) \ge \frac{1}{2}\gamma(G)\gamma(H) + \frac{1}{2}\min(\gamma(G),\gamma(H))$. We further improve their result by showing $\gamma(G \square H) \ge \frac{1}{2}\gamma(G)\gamma(H) + \frac{1}{2}\max(\gamma(G),\gamma(H)).$