The 2-rainbow domination number of Cartesian product of cycles

Abstract

A $k$-rainbow dominating function ($k$RDF) of $G$ is a function that assigns subsets of $\{1,2,...,k\}$ to the vertices of $G$ such that for vertices $v$ with $f(v)=\varnothing$ we have ${\bigcup }_{u\in N(v)}f(u)=\{1,2,...,k\}$. The weight $w(f)$ of a $k$RDF $f$ is defined as $w(f)={\sum }_{v\in V(G)}\left|f(v)\right|$. The minimum weight of a $k$RDF of $G$ is called the $k$-rainbow domination number of $G$, which is denoted by ${\gamma }_{rk}(G)$. In this paper, we study the 2-rainbow domination number of the Cartesian product of two cycles. Exact values are given for a number of infinite families and we prove lower and upper bounds for all other cases.