On a Vizing-type integer domination conjecture

TL;DR

This paper proves a Vizing-type conjecture for the $ ext{domination}$ number in graph theory, specifically for claw-free graphs using the Roman $ ext{2}$-domination concept, confirming the inequality for all $k ext{ } extgreater 1$.

Contribution

It establishes the conjecture for claw-free graphs using Roman $ ext{2}$-domination, extending the validity to all $k extgreater 1$ with a new proof approach.

Findings

Proves the inequality $ ext{γ}_{ ext{R2}}(G oxtimes H) extgreater= ext{γ}(G) ext{γ}(H)$ for claw-free graphs.

Confirms the Vizing-type conjecture for all $k extgreater 1$.

Uses Roman $ ext{2}$-domination to establish bounds in graph products.

Abstract

Given a simple graph $G$, a dominating set in $G$ is a set of vertices $S$ such that every vertex not in $S$ has a neighbor in $S$. Denote the domination number, which is the size of any minimum dominating set of $G$, by $\gamma(G)$. For any integer $k\ge 1$, a function $f : V (G) \rightarrow \{0, 1, . . ., k\}$ is called a \emph{$\{k\}$-dominating function} if the sum of its function values over any closed neighborhood is at least $k$. The weight of a $\{k\}$-dominating function is the sum of its values over all the vertices. The $\{k\}$-domination number of $G$, $\gamma_{\{k\}}(G)$, is defined to be the minimum weight taken over all $\{k\}$-domination functions. Bre\v{s}ar, Henning, and Klav\v{z}ar (On integer domination in graphs and Vizing-like problems. \emph{Taiwanese J. Math.} {10(5)} (2006) pp. 1317--1328) asked whether there exists an integer $k\ge 2$ so that $\gamma_{\{k\}}(G\square H)\ge \gamma(G)\gamma(H)$. In this note we use the Roman $\{2\}$-domination number, $\gamma_{R2}$ of Chellali, Haynes, Hedetniemi, and McRae, (Roman $\{2\}$-domination. \emph{Discrete Applied Mathematics} {204} (2016) pp. 22-28.) to prove that if $G$ is a claw-free graph and $H$ is an arbitrary graph, then $\gamma_{\{2\}}(G\square H)\ge \gamma_{R2}(G\square H)\ge \gamma(G)\gamma(H)$, which also implies the conjecture for all $k\ge 2$.