A note on Vizing's conjecture

TL;DR

This paper proves a special case of Vizing's conjecture relating the domination number of a Cartesian product of graphs to the domination numbers of the individual graphs, under specific conditions on dominating sets.

Contribution

It establishes a new partial result for Vizing's conjecture by identifying conditions under which the conjecture holds for the Cartesian product of graphs.

Findings

Proves a special case of Vizing's conjecture.

Shows that under certain domination set conditions, the conjecture's inequality holds.

Provides insight into the structure of dominating sets in graph products.

Abstract

Let $\gamma(G)$ denote the domination number of graph $G$. Let $G$ and $H$ be graphs and $G\Box H$ their Cartesian product. For $h\in V(H)$ define $G_h=\{(g,h)\,|\,g\in V(G)\}$ and call this set a $G$-layer of $G\Box H$. We prove the following special case of Vizing's conjecture. Let $D$ be a dominating set of $G\Box H$. If there exist minimum dominating sets $D_1$ and $D_2$ of $G$ such that for every $h\in V(H)$, the projection of $D\cap G_h$ to $G$ is contained in $D_1$ or $D_2$, then $|D|\geq \gamma(G)\gamma(H)$.