arXiv 1503.083712015-03math.CO

将$H$ -次自由图划分为无大分支的三个子图

Partitioning $H$-minor free graphs into three subgraphs with no large components

Chun-Hung Liu, Sang-il Oum

发表信息：: J. Combin. Theory, Ser. B, 128(January 2018), pp. 114-133
DOI：: 10.1016/j.jctb.2017.08.003

摘要：

本文证明了：对于任意的图$H$，如果一个图$H$没有（奇数）$H$子图，那么它的顶点集$V(G)$可以划分为三个集合$X_1$，$X_2$，$X_3$，使得对于每个$i$，在$X_i$上导出的子图没有大于$H$和$G$的最大度的函数的分支.这改进了Alon，Ding，Oporowski和Vertigan~（2003）的先前结果，即如果$G$没有$H$子式，则$V(G)$可以被划分为四个这样的集合。我们的定理推广了Esperet和Jaret ~（2014）的一个结果，他们证明了它对于可嵌入固定曲面上的图，并询问它是否对没有$H$子式的图成立。作为推论，我们证明了：对于每一个正整数$t$，如果图$G$没有$K_{t+1}$子图，那么它的顶点集$V(G)$可以被划分成$3t$集$X_1,\ldots,X_{3t}$，使得对于每一个~ $i$，在$X_i$上导出的子图没有大于~ $t$的一个函数的分支.这个推论改进了Wood~（2010）的一个结果，该结果指出$V(G)$可以划分成$\lceil 3.5t+2\rceil$这样的集合。

英文摘要：

We prove that for every graph $H$, if a graph $G$ has no (odd) $H$ minor, then its vertex set $V(G)$ can be partitioned into three sets $X_1$, $X_2$, $X_3$ such that for each~$i$, the subgraph induced on $X_i$ has no component of size larger than a function of~$H$ and the maximum degree of~$G$. This improves a previous result of Alon, Ding, Oporowski and Vertigan~(2003) stating that $V(G)$ can be partitioned into four such sets if $G$ has no $H$ minor. Our theorem generalizes a result of Esperet and Joret~(2014), who proved it for graphs embeddable on a fixed surface and asked whether it is true for graphs with no $H$ minor. As a corollary, we prove that for every positive integer $t$, if a graph $G$ has no $K_{t+1}$ minor, then its vertex set $V(G)$ can be partitioned into $3t$ sets $X_1,\ldots,X_{3t}$ such that for each~$i$, the subgraph induced on $X_i$ has no component of size larger than a function of~$t$. This corollary improves a result of Wood~(2010), which states that $V(G)$ can be partitioned into $\lceil 3.5t+2\rceil$ such sets.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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