An improved upper bound on the oriented diameter of graphs with diameter $4$

Let $f(d)$ be the smallest value for which every bridgeless graph $G$ with diameter $d$ admits a strong orientation $\overrightarrow{G}$ such that the diameter of $\overrightarrow{G}$ is at most $f(d)$. Chv\'atal and Thomassen (1978) established general bounds for $f(d)$ which implies $f(4)\geq 12$, and proved $f(2)=6$. Kwok et al. (2010) proved that $9\leq f(3)\leq 11$. Wang and Chen (2022) determined $f(3)=9$. Babu et al. (2021) showed $f(4)\leq 21$. In this paper, we improve the upper bound of $f(4)$ to $18$.