平面三角剖分的定向直径的改进界
Improved bounds on the oriented diameter of planar triangulations
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中文总结 AI 辅助
本文针对平面三角剖分的定向直径问题，改进了通用上界的首项常数，并给出4-连通平面三角剖分的定向直径新上界。
中文摘要 AI 辅助
连通无桥图$G$的定向直径，记为$\boldsymbol{\text{diam}}(G)$，是$G$的所有强定向中直径的最小值。我们研究平面三角剖分的定向直径，证明对于任意$n$个顶点的平面三角剖分$G$，有$\boldsymbol{\text{diam}}(G)\boldsymbol{\frac{2n+44}{5}}$。这将Ge、Liu和Wang之前给出的最优通用上界$\boldsymbol{\frac{n}{2}}$中的首项常数从$\frac{1}{2}$改进为$\frac{2}{5}$。我们还证明，每个$n$个顶点的4-连通平面三角剖分满足$\boldsymbol{\text{diam}}(G)\boldsymbol{\frac{n+17}{3}}$。
英文摘要
The oriented diameter of a connected bridgeless graph $G$, denoted by $\overrightarrow{\operatorname{diam}}(G)$, is the minimum diameter among all strong orientations of $G$. We study the oriented diameter of planar triangulations, and show that $\overrightarrow{\operatorname{diam}}(G)\leq \frac{2n+44}{5}$ 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 $1/2$ to $2/5$. We also prove that every $n$-vertex $4$-connected planar triangulation satisfies $\overrightarrow{\operatorname{diam}}(G)\leq \frac{n+17}{3}$.
