Vizing's conjecture: a two-thirds bound for claw-free graphs

TL;DR

This paper proves that for claw-free graphs, the domination number of their Cartesian product is at least two-thirds of the product of their individual domination numbers, advancing understanding of Vizing's conjecture.

Contribution

It establishes a new lower bound of two-thirds for the domination number in the Cartesian product of claw-free graphs, improving previous bounds.

Findings

The domination number of the Cartesian product of claw-free graphs is at least two-thirds of the product of their domination numbers.

Provides a significant bound related to Vizing's conjecture for a specific class of graphs.

Enhances theoretical understanding of domination in graph products.

Abstract

We show that for any claw-free graph $G$ and any graph $H$, $\gamma(G\square H)\geq \frac{2}{3}\gamma(G)\gamma(H)$, where $\gamma(G)$ is the domination number of $G$.