平面图的逆直径
Inversion Diameter of Planar Graphs
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中文总结 AI 辅助
该研究针对平面图，证明了其逆直径的严格上界，将一般平面图的逆直径上界从12优化至8，还给出了围长至少为4、5、6的平面图对应的更紧上界7、5、4。
中文摘要 AI 辅助
给定定向图$\boldsymbol{\rightarrow}{G}$和顶点子集$X \boldsymbol{\boldsymbol{\rightarrow}} V(\boldsymbol{\rightarrow}{G})$，$X$的逆操作是反转所有两端都在$X$中的弧的方向。对于简单图$G$，逆直径$\text{diam}(I(G))$是顶点集的逆操作下，$G$的两个定向之间的最大距离。我们证明了严格界$\text{diam}(I(G))\boldsymbol{\boldsymbol{\text{2χ}}}_a(G)-2$，其中$\boldsymbol{\text{χ}}_a(G)$是无环色数。因此，每个平面图的逆直径至多为8，改进了之前已知的界12。利用强退化论证，我们还分别得到围长至少为4、5、6的平面图的上界7、5和4。
英文摘要
Given an oriented graph $\vec{G}$ and a subset of vertices $X \subseteq V(\vec{G})$, the \emph{inversion} of $X$ is the operation that reverses the orientation of every arc with both endpoints in $X$. For a simple graph $G$, the inversion diameter $\operatorname{diam}(I(G))$ is the maximum distance between two orientations of $G$ under inversions of vertex sets. We prove the sharp bound \[ \operatorname{diam}(I(G))\le 2χ_a(G)-2, \] where $χ_a(G)$ is the acyclic chromatic number. Consequently, every planar graph has inversion diameter at most $8$, improving the previously known bound $12$. Using strong-degeneracy arguments, we also obtain upper bounds $7$, $5$, and $4$ for planar graphs of girth at least $4$, $5$, and $6$, respectively.
