Improved bounds on the oriented diameter of planar triangulations

The oriented diameter of a connected bridgeless graph $G$, denoted by $\stackrel{}{\mathrm{diam}}(G)$, is the minimum diameter among all strong orientations of $G$. We study the oriented diameter of planar triangulations, and show that $\stackrel{}{\mathrm{diam}}(G)\le \frac{2n+44}{5}$ for any $n$-vertex planar triangulation $G$. This improves the leading constant in the previous best general upper bound $\lceil \frac{n}{2}\rceil$, due to Ge, Liu, and Wang, from $1\mathrm{/}2$ to $2\mathrm{/}5$. We also prove that every $n$-vertex $4$-connected planar triangulation satisfies $\stackrel{}{\mathrm{diam}}(G)\le \frac{n+17}{3}$.