直径为 $4$ 的图的有向直径的改进上界
An improved upper bound for oriented diameter of graphs with diameter $4$
- Nankai University（南开大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出序贯定向框架与势函数分析，将直径为4的无桥图的有向直径上界从18改进至16，方法可推广至更大直径情形。
AI中文摘要：
设 $f(d)$ 表示最小的整数，使得每个直径为 $d$ 的无桥图都有一个直径至多为 $f(d)$ 的强定向。已知 $f(2)=6$ 且 $f(3)=9$。对于 $d=4$，Chvátal 和 Thomassen [JCTB, 1978] 的经典界给出 $12\le f(4)\le40$，后续工作将上界降至 21。最近，Lin、Wang 和 You 进一步建立了更强的界 $f(4)\le18$。将这个界降至 18 以下被证明是相当困难的，因为剩余的极值构型无法通过基于 $R-S$ 定向及相关局部构造的现有技术处理。在本文中，我们证明 $f(4)\le16$。我们的方法完全不同于以往的方法。我们不直接构造强定向，而是发展了一个序贯定向框架，辅以辅助距离函数和势函数分析。这使我们能够在全局上控制有向距离，同时避免早期方法所需的复杂情形分析。我们相信，这里引入的框架可能对研究更大直径的有向直径问题有用。
英文摘要：
Let $f(d)$ denote the smallest integer such that every bridgeless graph of diameter $d$ admits a strong orientation with diameter at most $f(d)$. It is known that $f(2)=6$ and $f(3)=9$. For $d=4$, the classical bounds of Chvátal and Thomassen [JCTB, 1978] imply $12\le f(4)\le40$, and subsequent work reduced the upper bound to 21. Very recently, Lin, Wang and You further established the substantially stronger bound $f(4)\le18$. Pushing this bound below $18$ turns out to be considerably more difficult, since the remaining extremal configurations cannot be handled by existing techniques based on $R-S$ orientations and related local constructions. In this paper, we prove that $f(4)\le16$. Our approach is entirely different from previous ones. Instead of constructing a strong orientation directly, we develop a sequential orientation framework together with auxiliary distance functions and a potential-function analysis. This enables us to control directed distances globally while avoiding the intricate case analysis required by earlier methods. We believe that the framework introduced here may be useful for studying oriented diameter problems of larger diameter.
