On a Vizing-type integer domination conjecture

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 $γ(G)$. 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 $G$, $γ_{\{k\}}(G)$, is defined to be the minimum weight taken over all $\{k\}$-domination functions. Brešar, Henning, and Klavžar (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 $γ_{\{k\}}(G\square H)\ge γ(G)γ(H)$. In this note we use the Roman $\{2\}$-domination number, $γ_{R2}$ 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 $γ_{\{2\}}(G\square H)\ge γ_{R2}(G\square H)\ge γ(G)γ(H)$, which also implies the conjecture for all $k\ge 2$.

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