arXiv 2005.022502020-05math.CO

齐次集、临界分离器、临界图和最优$χ$ -约束函数

Homogeneous sets, clique-separators, critical graphs, and optimal $χ$-binding functions

Christoph Brause, Maximilian Geißer, Ingo Schiermeyer

摘要：

给定一个图的集合$\mathcal{H}$，设$f_\mathcal{H}^\star\colon \mathbb{N}_{>0}\to \mathbb{N}_{>0}$是$\mathcal{H}$ -free图类的最优$χ$ -binding函数，即$$f_\mathcal{H}^\star(ω)=\max\{χ(G): G\text{ is } \mathcal{H}\text{-free, } ω(G)=ω\}.$$.本文将齐次集分解法和连通-分离器分解法相结合，以确定$P_5$ -free图和$(C_5,C_7,\ldots)$ -free图的子类的最优$χ$ -binding函数.特别是，我们证明了以下每个$ω\geq 1$：（i）$\ f_{\{P_5,banner\}}^\star(ω)=f_{3K_1}^\star(ω)\in Θ(ω^2/\log(ω)),$（ii）$\ f_{\{P_5,co-banner\}}^\star(ω)=f^\star_{\{2K_2\}}(ω)\in\mathcal{O}(ω^2),$（iii）$\ f_{\{C_5,C_7,\ldots,banner\}}^\star(ω)=f^\star_{\{C_5,3K_1\}}(ω)\notin \mathcal{O}(ω),$和（iv）$\ f_{\{P_5,C_4\}}^\star(ω)=\lceil(5ω-1)/4\rceil.$我们还将，对于我们考虑的每个图类，所有图$G$与$χ(G)>χ(G-u)$为每个$u\in V(G)$。从这些结构的结果，我们可以证明里德猜想-有关色数，团数，最大程度的一个图-$(P_5,banner)$ -自由图。

英文摘要：

Given a set $\mathcal{H}$ of graphs, let $f_\mathcal{H}^\star\colon \mathbb{N}_{>0}\to \mathbb{N}_{>0}$ be the optimal $χ$-binding function of the class of $\mathcal{H}$-free graphs, that is, $$f_\mathcal{H}^\star(ω)=\max\{χ(G): G\text{ is } \mathcal{H}\text{-free, } ω(G)=ω\}.$$ In this paper, we combine the two decomposition methods by homogeneous sets and clique-separators in order to determine optimal $χ$-binding functions for subclasses of $P_5$-free graphs and of $(C_5,C_7,\ldots)$-free graphs. In particular, we prove the following for each $ω\geq 1$: (i) $\ f_{\{P_5,banner\}}^\star(ω)=f_{3K_1}^\star(ω)\in Θ(ω^2/\log(ω)),$ (ii) $\ f_{\{P_5,co-banner\}}^\star(ω)=f^\star_{\{2K_2\}}(ω)\in\mathcal{O}(ω^2),$ (iii) $\ f_{\{C_5,C_7,\ldots,banner\}}^\star(ω)=f^\star_{\{C_5,3K_1\}}(ω)\notin \mathcal{O}(ω),$ and (iv) $\ f_{\{P_5,C_4\}}^\star(ω)=\lceil(5ω-1)/4\rceil.$ We also characterise, for each of our considered graph classes, all graphs $G$ with $χ(G)>χ(G-u)$ for each $u\in V(G)$. From these structural results, we can prove Reed's conjecture -- relating chromatic number, clique number, and maximum degree of a graph -- for $(P_5,banner)$-free graphs.

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