An Improvement on Vizing's Conjecture

TL;DR

This paper improves a bound related to Vizing's conjecture by establishing a new inequality involving the domination number and Roman domination number of graph Cartesian products.

Contribution

It proves a tighter inequality, showing that the product of domination numbers is bounded above by the Roman domination number of the Cartesian product, refining previous bounds.

Findings

Established that mma(G)mma(H)  mma_R(Gb7H) for all simple graphs G, H.

Improved the known inequality from mma(G)mma(H)  2 mma(Gb7H).

Provided a new bound that narrows the gap in Vizing's conjecture context.

Abstract

Let $\gamma(G)$ denote the domination number of a graph $G$. A {\it Roman domination function} of a graph $G$ is a function $f: V\to\{0,1,2\}$ such that every vertex with 0 has a neighbor with 2. The {\it Roman domination number} $\gamma_R(G)$ is the minimum of $f(V(G))=\Sigma_{v\in V}f(v)$ over all such functions. Let $G\square H$ denote the Cartesian product of graphs $G$ and $H$. We prove that $\gamma(G)\gamma(H) \leq \gamma_R(G\square H)$ for all simple graphs $G$ and $H$, which is an improvement of $\gamma(G)\gamma(H) \leq 2\gamma(G\square H)$ given by Clark and Suen \cite{CS}, since $\gamma(G\square H)\leq \gamma_R(G\square H)\leq 2\gamma(G\square H)$.