An Improved Bound for Upper Domination of Cartesian Products of Graphs

TL;DR

This paper establishes a new lower bound for the upper domination number of the Cartesian product of two graphs, advancing understanding of domination parameters in graph theory.

Contribution

It proves a conjectured inequality relating the upper domination number of a Cartesian product to those of the component graphs, filling a gap in graph domination theory.

Findings

Proves a lower bound for upper domination of Cartesian products.

Validates a conjecture proposed by Brešar.

Enhances theoretical understanding of domination in graph products.

Abstract

In this paper, we prove a problem proposed by Bre\v{s}ar: for any graphs $G$ and $H$, $\Gamma(G\square H)\ge\Gamma(G)\Gamma(H)+ \min\{|V(G)|-\Gamma(G),|V(H)|-\Gamma(H)\}$, where $\Gamma(G)$ denotes the upper domination number of $G$.