基于完全可除性的无导出分叉的着色图类
Coloring graph classes with no induced fork via perfect divisibility
摘要：
对于图$G$，$χ(G)$表示它的色数，而$ω(G)$表示它的团数。一个图$G$被称为完全可分的，如果对于$G$的所有导出子图$H$，$V(H)$可以被划分为两个集合$A$，$B$，使得$H[A]$是完全的，并且$ω(H[B]) < ω(H)$ .一个整数值函数$f$称为遗传图类$\cal C$的$χ$ -结合函数，如果对每个图$G\in \cal C$都有$χ(G) \leq f(ω(G))$.叉是完全二部图$K_{1,3}$的一条边细分一次得到的图。找到一个多项式$χ$绑定功能的类fork-free图的问题是开放的。本文研究了几类无叉图的结构，特别是在完全可除性的背景下研究了无（fork，$F$）图类$\cal G$，其中$F$是一个稳定集为3的五点图，并证明了每个$G\in \cal G$都满足$χ(G)\leq ω(G)^2$ .我们还注意到类$\cal G$不允许线性$χ$绑定函数.
英文摘要：
For a graph $G$, $χ(G)$ will denote its chromatic number, and $ω(G)$ its clique number. A graph $G$ is said to be perfectly divisible if for all induced subgraphs $H$ of $G$, $V(H)$ can be partitioned into two sets $A$, $B$ such that $H[A]$ is perfect and $ω(H[B]) < ω(H)$. An integer-valued function $f$ is called a $χ$-binding function for a hereditary class of graphs $\cal C$ if $χ(G) \leq f(ω(G))$ for every graph $G\in \cal C$. The fork is the graph obtained from the complete bipartite graph $K_{1,3}$ by subdividing an edge once. The problem of finding a polynomial $χ$-binding function for the class of fork-free graphs is open. In this paper, we study the structure of some classes of fork-free graphs; in particular, we study the class of (fork,$F$)-free graphs $\cal G$ in the context of perfect divisibility, where $F$ is a graph on five vertices with a stable set of size three, and show that every $G\in \cal G$ satisfies $χ(G)\leq ω(G)^2$. We also note that the class $\cal G$ does not admit a linear $χ$-binding function.
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