发表机构
Nankai University; The Hong Kong Polytechnic University（南开大学; 香港理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了连通无桥图的最大定向直径$f(d)$满足$f(d)\le \tfrac12d^2+7d$，匹配已知下界的前导系数，确立了最优二次系数，并给出多项式时间算法构造满足界的强定向。
AI 中文摘要
连通无桥图的定向直径是其强定向的最小直径。设$f(d)$为所有直径为$d$的此类图中最大的定向直径。1978年，Chvátal和Thomassen证明了$f(d)\le2d^2+2d$，并构造了图表明$f(d)$的任何二次上界的前导系数至少为$1/2$。我们证明对于每个整数$d\ge1$，$f(d)\le \tfrac12d^2+7d$。这与他们的下界的前导系数匹配，并确立了$f(d)=\tfrac12d^2+O(d)$，确定了最优二次系数。我们的证明给出了一个多项式时间算法，构造满足所述界的强定向。
英文摘要
The oriented diameter of a connected bridgeless graph is the minimum diameter of a strong orientation. Let $f(d)$ be the maximum oriented diameter among all such graphs of diameter $d$. In 1978, Chvátal and Thomassen proved $f(d)\le2d^2+2d$ and constructed graphs showing that any quadratic upper bound on $f(d)$ must have leading coefficient at least $1/2$. We prove that $f(d)\le \tfrac12d^2+7d$ for every integer $d\ge1$. This matches the leading coefficient of their lower bound and establishes $f(d)=\tfrac12d^2+O(d)$, determining the optimal quadratic coefficient. Our proof gives a polynomial-time algorithm that constructs a strong orientation satisfying the stated bound.
