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导出子图与树分解 II：有界度图中墙及其线图的研究

Induced subgraphs and tree decompositions II. Toward walls and their line graphs in graphs of bounded degree

Tara Abrishami, Maria Chudnovsky, Cemil Dibek, Sepehr Hajebi, Paweł Rzążewski, Sophie Spirkl, Kristina Vušković

备注：
Accepted manuscript; see DOI for journal version
发表信息：
Journal of Combinatorial Theory, Series B, Volume 164, January 2024, Pages 371-403
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AI总结：

本文研究大树宽有界度图的不可避免导出子图，证明了两个支持Aboulker等人猜想的定理：一是大树宽有界度图必含t-theta、t-pyramid或墙细分线图作为导出子图，二是必含给定次立方细分毛毛虫或其细分线图作为导出子图。

摘要：

本文的动机源于以下问题：大树宽图的不可避免导出子图是什么？Aboulker等人提出了一个猜想，在有界最大度图中回答了这个问题，断言对于所有$k$和$Δ$，每个最大度至多为$Δ$且树宽足够大的图，要么包含$(k\times k)$-墙的一个细分，要么包含$(k\times k)$-墙的一个细分的线图作为导出子图。我们证明了两个支持该猜想的定理，如下所述。\n1. 对于$t\geq 2$，一个$t$-theta是由两个不相邻顶点和它们之间三条内部不相交的路径组成的图，每条路径长度至少为$t$。一个$t$-pyramid是由一个顶点$v$、一个与$v$不相交的三角形$B$以及三条从$v$出发且其余部分不相交的路径组成的图，每条路径将$v$连接到$B$的一个顶点，且每条路径长度至少为$t$。我们证明对于所有$k,t$和$Δ$，每个最大度至多为$Δ$且树宽足够大的图，要么包含一个$t$-theta，要么包含一个$t$-pyramid，要么包含$(k\times k)$-墙的一个细分的线图作为导出子图。这肯定地回答了Pilipczuk等人提出的一个问题，该问题询问是否每个有界最大度且树宽足够大的图都包含一个theta或一个三角形作为导出子图（其中theta表示某个$t\geq 2$的$t$-theta）。\n2. 一个次立方细分毛毛虫是一棵最大度至多为三的树，其所有度为三的顶点都位于一条路径上。我们证明对于每个$Δ$和次立方细分毛毛虫$T$，每个最大度至多为$Δ$且树宽足够大的图，要么包含$T$的一个细分，要么包含$T$的一个细分的线图作为导出子图。

英文摘要：

This paper is motivated by the following question: what are the unavoidable induced subgraphs of graphs with large treewidth? Aboulker et al. made a conjecture which answers this question in graphs of bounded maximum degree, asserting that for all $k$ and $Δ$, every graph with maximum degree at most $Δ$ and sufficiently large treewidth contains either a subdivision of the $(k\times k)$-wall or the line graph of a subdivision of the $(k\times k)$-wall as an induced subgraph. We prove two theorems supporting this conjecture, as follows. 1. For $t\geq 2$, a $t$-theta is a graph consisting of two nonadjacent vertices and three internally disjoint paths between them, each of length at least $t$. A $t$-pyramid is a graph consisting of a vertex $v$, a triangle $B$ disjoint from $v$ and three paths starting at $v$ and disjoint otherwise, each joining $v$ to a vertex of $B$, and each of length at least $t$. We prove that for all $k,t$ and $Δ$, every graph with maximum degree at most $Δ$ and sufficiently large treewidth contains either a $t$-theta, or a $t$-pyramid, or the line graph of a subdivision of the $(k\times k)$-wall as an induced subgraph. This affirmatively answers a question of Pilipczuk et al. asking whether every graph of bounded maximum degree and sufficiently large treewidth contains either a theta or a triangle as an induced subgraph (where a theta means a $t$-theta for some $t\geq 2$). 2. A subcubic subdivided caterpillar is a tree of maximum degree at most three whose all vertices of degree three lie on a path. We prove that for every $Δ$ and subcubic subdivided caterpillar $T$, every graph with maximum degree at most $Δ$ and sufficiently large treewidth contains either a subdivision of $T$ or the line graph of a subdivision of $T$ as an induced subgraph.

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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