On 2-rainbow domination of some families of graphs

In this paper we find the $2-$rainbow domination number of the $4$-regular Harary graphs $H_{4,n}$, $ngeq 5$. Upper bound for the $2$-rainbow domination number of $P_1times P_m$ is found. At the end we find the lower bound for the $2$-rainbow domination number of $k$-regular graphs.