Bounds on the 2-rainbow domination number of graphs

Abstract

A {\it 2-rainbow domination function} of a graph $G$ is a function $f$ that assigns to each vertex a set of colors chosen from the set $\{1,2\}$, such that for any $v\in V(G)$, $f(v)=\varnothing$ implies ${\bigcup }_{u\in N(v)}f(u)=\{1,2\}$. The {\it 2-rainbow domination number ${\gamma }_{r2}(G)$} of a graph $G$ is the minimum $w(f)={\Sigma }_{v\in V}|f(v)|$ over all such functions $f$. Let $G$ be a connected graph of order $|V(G)|=n\ge 3$. We prove that ${\gamma }_{r2}(G)\le 3n/4$ and we characterize the graphs achieving equality. We also prove a lower bound for 2-rainbow domination number of a tree using its domination number. Some other lower and upper bounds of ${\gamma }_{r2}(G)$ in terms of diameter are also given.

References in corpus (1)

• An Improvement on Vizing's Conjecture