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 $f(d)$. Chvátal and Thomassen (JCTB, 1978) established general bounds for $f(d)$, and also proved that $f(2)=6$ and $f(4)\ge 12$. The works of both Kwok, Liu and West (JCTB, 2010) and Wang and Chen (JCTB, 2022) together determined $f(3)=9$. In this paper, we improve the best known upper bound for $f(4)$ from $21$ (Babu et al., DAM, 2021) to $18$.