A double Roman Dominating function on a graph $G$ is a function $ f:V\rightarrow \{ 0,1,2,3\}$ such that the following conditions hold. If $f(v)=0$, then vertex $v$ must have at least two neighbors in $V_2$ or one neighbor in $V_3$ and if $f(v)=1$, then vertex $v$ must have at least one neighbor in $V_2\bigcup V_3$. The weight of a double Roman dominating function is the sum $w_f=\sum_{ v\in V(G)}{ f(v)}$. In this paper, we improve the upper bounds of $\gamma_{ dR}(G)$ that has already obtained and we show that $\gamma_{ dR}(G)\leq\dfrac{ 12n}{ 11}$, for any graph with $\delta(G) \ge 2$. This bound improve the bounds that have already been presented in \cite{ chen} and \cite{ kkcs}. Finally we prove the conjecture posed in \cite{ kkcs}.
Categories: math.CO