极大（外）平面图的拉姆齐性质

中文摘要 AI 辅助

我们研究相对于极大平面图类和极大外平面图类的拉姆齐理论的自然扩展，这可视为Axenovich等人引入的“平面图拉姆齐理论”研究的延续。我们提出的问题是：对于固定的图族$\boldsymbol{\textit{K}}$以及一对图$\boldsymbol{\textit{\textbraceleft H,F\textbraceright}}$，是否存在整数$\boldsymbol{r_{\textit{K}} (H, F)}$，使得对每个满足$\boldsymbol{|G| \boldsymbol{\textgreater} r_{\textit{K}}(H, F)}$的图$\boldsymbol{G \boldsymbol{\textin \textit{K}}}$，对$G$的每条边进行红/蓝着色时，必然存在红色的$H$副本或蓝色的$F$副本？当这样的整数存在时，我们称$\boldsymbol{\textbraceleft H,F\textbraceright}$在$\boldsymbol{\textit{K}}$中是不可避免的，否则称其在$\boldsymbol{\textit{K}}$中是可避免的。本研究聚焦于$\boldsymbol{\textit{K = \textit{K}_{\text{MOP}}}}$和$\boldsymbol{\textit{K = \textit{K}_{\text{MP}}}}$的情况，其中$\boldsymbol{\textit{K}_{\text{MOP}}}$表示极大外平面图（MOP）族，$\boldsymbol{\textit{K}_{\text{MP}}}$表示极大平面图（MP）族。该框架概括了相对于这些图族的经典拉姆齐问题，因为当$\boldsymbol{\textit{K = \textbraceleft K_n \boldsymbol{:} n \boldsymbol{\textgreater} 2\textbraceright}}$时，对应经典拉姆齐问题。我们还研究了MOP和MP对应的拉姆齐数，记为$\boldsymbol{r_{\text{MOP}}(H, F)}$和$\boldsymbol{r_{\text{MP}}(H, F)}$。当$\boldsymbol{\textit{K = \textit{K}_{\text{MOP}}}}$时，我们完全确定了所有满足$\boldsymbol{|E(F)| \boldsymbol{\textgreater} 2}$的不可避免对$\boldsymbol{\textbraceleft H, F\textbraceright}}$，以及$\boldsymbol{r_{\text{MOP}}(H, F)}$的上界，部分情况下还得到了精确值。当$\boldsymbol{\textit{K = \textit{K}_{\text{MP}}}}$时，我们完全确定了连通图$\boldsymbol{H}$的对角情况$\boldsymbol{\textbraceleft H, H\textbraceright}}$的所有不可避免对，证明$H$必须是图$\boldsymbol{P_3}$、$\boldsymbol{P_4}$、$\boldsymbol{P_5}$、$\boldsymbol{K_{1,3}}$或叉图$\boldsymbol{S_{2,1,1}}$之一。本研究为相对于图族的拉姆齐理论研究开辟了更多可能性，并提出了若干相关的开放问题。

英文摘要

We study a natural extension of Ramsey theory relative to the classes of maximally planar and maximally outerplanar graphs. This can be seen as a continuation of the study of `Planar Ramsey theory', introduced by Axenovich et al. The question we ask is the following: For a fixed family $\mathcal{K}$ of graphs and a pair of graphs $\{H,F\}$, does there exist an integer $r_{\mathcal{K}} (H, F)$ such that for every graph $G \in \mathcal{K}$ with $|G| \geq r_{\mathcal{K}}(H, F)$, every red/blue edge-colouring of $G$ admits a red copy of $H$ or a blue copy of $F$? When such an integer exists, we say $\{H,F\}$ is unavoidable in $\mathcal{K}$,, and otherwise $\{H,F\}$ is avoidable in $\mathcal{K}$. Our work focuses on this problem where $\mathcal{K} = \mathcal{K}_{\mathrm{MOP}}$ and $\mathcal{K} = \mathcal{K}_{\mathrm{MP}}$, which denote the families of maximal outerplanar (MOP) graphs and maximal planar (MP) graphs, respectively. This framework generalises the classical Ramsey problem relative to these classes, as the case with $\mathcal{K} = \{K_n \colon n \geq 2\}$ corresponds to classical Ramsey. We also study the corresponding Ramsey numbers for MOP and MP, which we denote as $r_{\mathrm{MOP}}(H, F)$ and $r_{\mathrm{MP}}(H, F)$. In the case when $\mathcal{K} = \mathcal{K}_{\mathrm{MOP}}$, we completely determine all unavoidable pairs $\{H, F\}$ with $|E(F)| \geq 2$, together with upper bounds and sometimes exact values of $r_{\mathrm{MOP}}(H, F)$. When $\mathcal{K} = \mathcal{K}_{\mathrm{MP}}$, we completely determine all unavoidable pairs in the diagonal case $\{H, H\}$ when $H$ is connected, showing that $H$ must be one of the graphs $P_3$, $P_4$, $P_5$, $K_{1, 3}$ or the fork graph $S_{2,1,1}$. This work opens up further possibilities in the study of Ramsey theory relative to a class, and we offer several open problems in this vein.

发表机构

• University of Birmingham（伯明翰大学）
• University of Haifa-Oranim（海法-奥拉宁大学）
• University of Oxford（牛津大学）

机构由 AI 辅助整理，请以论文原文为准。

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24 pages, 19 figures