A new bound for Vizing's conjecture

TL;DR

This paper introduces a new bound for Vizing's conjecture on the domination number of graph Cartesian products, providing improved inequalities and a simplified proof for the case when the domination number is three.

Contribution

It presents a novel bound involving the power of a graph, extends the bound to specific cases based on the parity of the domination number, and offers a concise proof for the case .

Findings

Established a new lower bound for 's conjecture.

Derived bounds for graphs with odd and even domination numbers.

Provided a simplified proof for the case .

Abstract

For any graph $G$, we define the power $\pi(G)$ as the minimum of the largest number of neighbors in a $\gamma$-set of $G$, of any vertex, taken over all $\gamma$-sets of $G$. We show that $\gamma(G\square H)\geq \frac{\pi(G)}{2\pi(G) -1}\gamma(G)\gamma(H)$. Our methods allow us to prove the following statements for any graphs $G$ and $H$, (1) $\gamma(G\square H)\geq \frac{\lceil \frac{\gamma (G)}{2}\rceil}{2\lceil \frac{\gamma (G)}{2}\rceil-1}\gamma(G)\gamma(H)$ for odd $\gamma(G)$, (2) $\gamma(G\square H)\geq \frac{\gamma (G)}{2\gamma (G)-2}\gamma(G)\gamma(H)$, for even $\gamma(G)$, and (3) a short proof of Vizing's conjecture where $\gamma(G)=3$. Our argument relies on establishing efficient correspondences between dominating vertices and subsets of their neighborhoods and then showing a sufficient number of dominating vertices that horizontally dominate vertically undominated cells.