For a graph $G$, $\chi(G)$ will denote its chromatic number, and $\omega(G)$ 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 $\omega(H[B]) < \omega(H)$. An integer-valued function $f$ is called a $\chi$-binding function for a hereditary class of graphs $\cal C$ if $\chi(G) \leq f(\omega(G))$ 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 $\chi$-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 $\chi(G)\leq \omega(G)^2$. We also note that the class $\cal G$ does not admit a linear $\chi$-binding function.

中文翻译：

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通过完美的除法为图类着色，而不会引起分叉

对于图$ G $，$ \ chi（G）$将表示其色数，而$ \ omega（G）$将表示其色数。如果对于所有诱导子图$ G $的$ H $，$ V（H）$可以划分为两组$ A $，$ B $这样$ H [A]，则说图$ G $是完全可整的$是完美的，而$ \ omega（H [B]）<\ omega（H）$。如果每个图的$ \ chi（G）\ leq f（\ omega（G））$，图的遗传类$ \ cal C $的整数值函数$ f $称为$ \ chi $绑定函数$ G \ in \ cal C $。叉子是从完整的二分图$ K_ {1,3} $通过细分一次边获得的图。寻找无叉图类的多项式$ \ chi $ -binding函数的问题是开放的。在本文中，我们研究了一些无叉图类的结构。特别是，我们在完全可除的情况下研究了（叉，$ F $）无图$ \ cal G $的类别，其中$ F $是五个顶点的图形，具有五个稳定的大小集，并显示\ cal G $中的每个$ G \ in \ cal G $都满足$ \ chi（G）\ leq \ omega（G）^ 2 $。我们还注意到，类$ \ cal G $不接受线性$ \ chi $绑定函数。