On the Domination Number of the Product of Two Cycles

Abstract

Let $G=(V,E)$ be a graph. A subset $D\subseteq V$ is called a dominating set for $G$ if for every $v\in V–D$, $v$ is adjacent to some vertex in $D$. The domination number $\gamma (G)$ is equal to $min\{|D|:D\text{ is a dominating set of }G\}$.

In this paper, we calculate the domination numbers $\gamma (C_m\times C_n)$ of the product of two cycles $C_m$ and $C_n$ of lengths $m$ and $n$ for $m=5$ and $n=3\mathrm{mod}5$, also for $m=6,7$ and arbitrary $n$.