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Let $G=(V,E)$ be a graph of order $n$ and let $\gamma _{R}(G)$ and $\partial(G)$ denote the Roman domination number and the differential of $G,$respectively. In this paper we prove that for any integer $k\geq 0$, if $G$ isa graph of order $n\geq 6k+9$, minimum degree $\delta \geq 2,$ which does notcontain any induced $\{C_{5},C_{8},\ldots ,C_{3k+2}\}$% -cycles, then $\gamma_{R}(G)\leq \frac{(4k+8)n}{6k+11}$. This bound is an improvement of the boundsgiven in [E.W. Chambers, B. Kinnersley, N. Prince, and D.B. West, Extremalproblems for Roman domination, SIAM J. Discrete Math. 23 (2009) 1575--1586]when $k=0,$ {and [S. Bermudo, On the differential and Roman domination numberof a graph with minimum degree two, Discrete Appl. Math. 232 (2017), 64--72]when }$k=1.$ Moreover, using the Gallai-type result involving the Romandomination number and the differential of graphs established by Bermudo et al.stating that $\gamma _{R}(G)+\partial (G)=n$, we have $\partial (G)\geq\frac{(2k+3)n}{6k+11},$ thereby settling the conjecture of Bermudo posed in thesecond paper.

Roman domination in graphs with minimum degree at least two and some forbidden cycles