On the oriented diameter of graphs with given minimum degree

Erdős, Pach, Pollack, and Tuza [$\mathit{\text{J.}}\mathit{\text{Combin.}}\mathit{\text{Theory}}\mathit{\text{Ser.}}\mathit{\text{B,}}\mathit{\text{47(1)}}\mathit{\text{(1989),}}\mathit{\text{73-79}}$] proved that the diameter of a connected $n$-vertex graph with minimum degree $\delta$ is at most $\frac{3n}{\delta +1}+O(1)$. The oriented diameter of an undirected graph $G$, denoted by $\overrightarrow{\text{diam}}(G)$, is the minimum diameter of a strongly connected orientation of $G$. Bau and Dankelmann [$\mathit{\text{European}}\mathit{\text{J.}}\mathit{\text{Combin.,}}\mathit{\text{49}}\mathit{\text{(2015),}}\mathit{\text{126-133}}$] showed that for every bridgeless $n$-vertex graph $G$ with minimum degree $\delta$, $\overrightarrow{\text{diam}}(G)\le \frac{11n}{\delta +1}+9$. They also showed an infinite family of graphs with oriented diameter at least $\frac{3n}{\delta +1}+O(1)$ and posed the problem of determining the smallest possible value $c$ for which $\overrightarrow{\text{diam}}(G)\le c\cdot \frac{3n}{\delta +1}+O(1)$ holds. In this paper, we show that the smallest value $c$ such that the upper bound above holds for all $\delta \ge 2$ is $1$, which is best possible.