发表机构
Nankai University; Nanjing University; The Hong Kong Polytechnic University; East China Normal University; Fuzhou University（南开大学; 南京大学; 香港理工大学; 华东师范大学; 福州大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了有向直径函数$f(d)$满足$f(d)=\lceil d^2/2\rceil+d+O(1)$，通过构造中心子图并联合估计路径，将上下界差缩小到常数。
AI 中文摘要
设$f(d)$为最小的整数，使得每个直径为$d$的有限连通无桥简单图都有一个直径至多为$f(d)$的强定向。Chvátal和Thomassen（1978）证明了$\lceil d^2/2\rceil+d\le f(d)\le2d^2+2d$。本文证明了对每个$d\ge2$，有$\lceil d^2/2\rceil+d\le f(d)\le\lceil d^2/2\rceil+d+18$，这表明$f(d)=\lceil d^2/2\rceil+d+O(1)$，并确定了二次项和线性项至多相差一个有界加性误差。我们证明的关键方法是构造一个中心子图$H$，它允许直径为$O(d)$的强定向，并且与外部每个顶点的距离至多为$\lfloor d/2\rfloor$。我们通过在实际附着顶点处联合估计外部路径及其在$H$中的连接路径，并在必要时重建$H$，从而获得精确的线性系数。
英文摘要
Let $f(d)$ be the smallest integer such that every finite connected bridgeless simple graph of diameter $d$ admits a strong orientation of diameter at most $f(d)$. Chvátal and Thomassen (1978) proved $\lceil d^2/2\rceil+d\le f(d)\le2d^2+2d$. In this paper, we prove that $\lceil d^2/2\rceil+d\le f(d)\le\lceil d^2/2\rceil+d+18$ for every $d\ge2$, which shows that $f(d)=\lceil d^2/2\rceil+d+O(1)$ and determines both the quadratic and linear terms up to a bounded additive error. The key method of our proof is to construct a central subgraph $H$ that admits a strong orientation of diameter $O(d)$ and is within distance $\lfloor d/2\rfloor$ of every vertex outside it. We obtain the sharp linear coefficient by jointly estimating outside paths and their connecting paths in $H$ at the actual attachment vertices, rebuilding $H$ when necessary.
Comments28 pages
