An asymptotically tight bound on oriented diameter

Abstract

Let $f(d)$ be the smallest integer such that every connected bridgeless graph of diameter $d$ admits an orientation with directed diameter at most $f(d)$. Chvátal and Thomassen (1978) proved $\lceil d^2/2\rceil +d\le f(d)\le 2d^2+2d$. In this paper, we prove that $f(d)\le \lceil d^2/2\rceil +6d$, which shows $f(d)=d^2/2+\Theta (d)$ and determine the leading quadratic coefficient of $f(d)$. The key method of our proof is to construct a central subgraph $H$ such that it admits a strong orientation of diameter $O(d)$ and the distance between $H$ and every vertex outside $H$ is at most $d/2$.