A class of graphs approaching Vizing's conjecture

TL;DR

This paper introduces classes of graphs approaching Vizing's conjecture, providing new bounds on the domination number of Cartesian product graphs for graphs in these classes.

Contribution

The paper defines classes of graphs _{n} and proves a new lower bound on the domination number for graphs in class _{1}, advancing understanding of Vizing's conjecture.

Findings

For graphs in class _{1}, ; ; .

Establishes a bound ; for ; graphs, improving previous bounds.

Class _{0} corresponds to class A of Bartsalkin and German.

Abstract

For any graph $G=(V,E)$, a subset $S\subseteq V$ \emph{dominates} $G$ if all vertices are contained in the closed neighborhood of $S$, that is $N[S]=V$. The minimum cardinality over all such $S$ is called the domination number, written $\gamma(G)$. In 1963, V.G. Vizing conjectured that $\gamma(G \square H) \geq \gamma(G)\gamma(H)$ where $\square$ stands for the Cartesian product of graphs. In this note, we define classes of graphs $\mathcal{A}_n$, for $n\geq 0$, so that every graph belongs to some such class, and $\mathcal{A}_0$ corresponds to class $A$ of Bartsalkin and German. We prove that for any graph $G$ in class $\mathcal{A}_1$, $\gamma(G\square H)\geq \left(\gamma(G)-\sqrt{\gamma(G)}\right)\gamma(H)$.