arXiv 1302.08122013-02-04math.CO

排除图对

Excluding Pairs of Graphs

Maria Chudnovsky, Alex Scott, Paul Seymour

摘要：

对于一个图$G$和一组图$\mathcal{H}$，如果$G$的导出子图都不同构于$\mathcal{H}$的一个成员，则称$G$是{ \em $\mathcal{H}$ -free}.给定一个整数$P>0$，一个图$G$和一组图$\mathcal{F}$，我们说$G$ { \em允许一个$(\mathcal{F},P)$ -划分}，如果$G$的顶点集可以划分成$P$子集$X_1,..., X_P$，使得对于每个$i \in \{1,..., P\}$，$|X_i|=1$或由$X_i$诱导的$G$的子图对于某些$F \in \mathcal{F}$是$\{F\}$ -free的.我们的第一个结果如下。对于每一对图$(H,J)$，使得$H$是两个图$H_1$和$H_2$的不交并，而$J$的补图$J^c$是两个图$J_1^c$和$J_2^c$的不交并，存在整数$P>0$使得每一个无$\{H,J\}$的图都有一个$(\{H_1,H_2,J_1,J_2\},P)$ -划分.使用类似的想法，我们也给出了一个简短的证明\cite{heroes}的结果之一。我们的最终结果是一个结构，表明如果$\{H,J\}$是图，每个至少有一个边缘，那么对于每一对整数$r,k$，存在一个图$G$，使得$G$的每个$r$ -顶点诱导子图是$\{H,J\}$ -分裂的，但$G$不允许$(\{H,J\},k)$ -划分.

英文摘要：

For a graph $G$ and a set of graphs $\mathcal{H}$, we say that $G$ is {\em $\mathcal{H}$-free} if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. Given an integer $P>0$, a graph $G$, and a set of graphs $\mathcal{F}$, we say that $G$ {\em admits an $(\mathcal{F},P)$-partition} if the vertex set of $G$ can be partitioned into $P$ subsets $X_1,..., X_P$, so that for every $i \in \{1,..., P\}$, either $|X_i|=1$, or the subgraph of $G$ induced by $X_i$ is $\{F\}$-free for some $F \in \mathcal{F}$. Our first result is the following. For every pair $(H,J)$ of graphs such that $H$ is the disjoint union of two graphs $H_1$ and $H_2$, and the complement $J^c$ of $J$ is the disjoint union of two graphs $J_1^c$ and $J_2^c$, there exists an integer $P>0$ such that every $\{H,J\}$-free graph has an $(\{H_1,H_2,J_1,J_2\},P)$-partition. Using a similar idea we also give a short proof of one of the results of \cite{heroes}. Our final result is a construction showing that if $\{H,J\}$ are graphs each with at least one edge, then for every pair of integers $r,k$ there exists a graph $G$ such that every $r$-vertex induced subgraph of $G$ is $\{H,J\}$-split, but $G$ does not admits an $(\{H,J\},k)$-partition.

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。

摘要：

英文摘要：

相关论文