置换与$χ$ -有界性
Substitution and $χ$-Boundedness
摘要：
一个图类$\mathcal{G}$称为{ \em $χ$ -bounded}，如果存在一个函数$f:\mathbb{N} \rightarrow \mathbb{R}$使得对于所有的$G \in \mathcal{G}$和$G$的所有诱导子图$H$，$χ(H) \leq f(ω(H))$ .在本文中，我们证明了，如果$\mathcal{G}$是一个$χ$有界类，那么是封闭的$\mathcal{G}$下的任何一个以下三个操作：替代，沿团胶合，胶合沿有限数量的顶点。此外，如果$\mathcal{G}$是由多项式（分别为：指数）函数$χ$ -有界的，则$\mathcal{G}$在替换下的闭包也是由某个多项式（分别为：指数）函数$χ$ -有界的。此外，我们表明，如果$\mathcal{G}$是一个$χ$有界类，然后封闭的$\mathcal{G}$下的操作胶沿集团和胶沿有限数量的顶点在一起也是$χ$有界的，因为是封闭的$\mathcal{G}$下的操作替代和胶沿集团在一起。
英文摘要：
A class $\mathcal{G}$ of graphs is said to be {\em $χ$-bounded} if there is a function $f:\mathbb{N} \rightarrow \mathbb{R}$ such that for all $G \in \mathcal{G}$ and all induced subgraphs $H$ of $G$, $χ(H) \leq f(ω(H))$. In this paper, we show that if $\mathcal{G}$ is a $χ$-bounded class, then so is the closure of $\mathcal{G}$ under any one of the following three operations: substitution, gluing along a clique, and gluing along a bounded number of vertices. Furthermore, if $\mathcal{G}$ is $χ$-bounded by a polynomial (respectively: exponential) function, then the closure of $\mathcal{G}$ under substitution is also $χ$-bounded by some polynomial (respectively: exponential) function. In addition, we show that if $\mathcal{G}$ is a $χ$-bounded class, then the closure of $\mathcal{G}$ under the operations of gluing along a clique and gluing along a bounded number of vertices together is also $χ$-bounded, as is the closure of $\mathcal{G}$ under the operations of substitution and gluing along a clique together.
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