  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 $G$ is
a graph of order $n\geq 6k+9$, minimum degree $\delta \geq 2,$ which does not
contain 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 bounds
given in [E.W. Chambers, B. Kinnersley, N. Prince, and D.B. West, Extremal
problems for Roman domination, SIAM J. Discrete Math. 23 (2009) 1575--1586]
when $k=0,$ {and [S. Bermudo, On the differential and Roman domination number
of a graph with minimum degree two, Discrete Appl. Math. 232 (2017), 64--72]
when }$k=1.$ Moreover, using the Gallai-type result involving the Roman
domination 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 the
second paper.
