A note on the bounds of Roman domination numbers

Let $ G $ be a graph and $ f: V(G) \rightarrow \{0, 1, 2\} $ be a mapping. $ f $ is said to be a Roman dominating function of $ G $ if every vertex $ u $ for which $ f(u) = 0 $ is adjacent to at least one vertex $ v $ for which $ f(v) = 2 $. The weight $ w(f) $ of a Roman dominating function $ f $ is the value $ w(f) = \sum_{u \in V(G)}f(u) $, and the minimum weight of a Roman dominating function is the Roman domination number $ \gamma_R(G) $. $ f $ is said to be a Roman $ \{2\} $-dominating function of $ G $ if $ \sum_{v\in N(u)}f(v)\geq 2 $ for every vertex $ u $ with $ f(u) = 0 $, where $ N(u) $ is the set of neighbors of $ u $ in $ G $. The weight of a Roman {2}-dominating function $ f $ is the sum $ \sum_{v\in V}f(v) $ and the minimum weight of a Roman {2}-dominating function is the Roman {2}-domination number $ \gamma_{\{R2\}}(G) $. Chellali et al. (2016) proved that $ \gamma_{R}(G)\geq\frac{\Delta+1}{\Delta}\gamma(G) $ for every nontrivial connected graph $ G $ with maximum degree $ \Delta $. In this paper, we generalize this result on nontrivial connected graph $ G $ with maximum degree $ \Delta $ and minimum degree $ \delta $. We prove that $ \gamma_{R}(G)\geq\frac{\Delta+2\delta}{\Delta+\delta}\gamma(G) $, which also implies that $ \frac{3}{2}\gamma(G)\leq\gamma_{R}(G)\leq 2\gamma(G) $ for any nontrivial regular graph. Moreover, we prove that $ \gamma_{R}(G)\leq2\gamma_{\{R2\}}(G)-1 $ for every graph $ G $ and there exists a graph $ I_k $ such that $ \gamma_{\{R2\}}(I_k) = k $ and $ \gamma_{R}(I_k) = 2k-1 $ for any integer $ k\geq2 $.