{"title":"Pure Pairs. VIII. Excluding a Sparse Graph","authors":"Alex Scott, Paul Seymour, Sophie Spirkl","doi":"10.1007/s00493-024-00117-z","DOIUrl":null,"url":null,"abstract":"<p>A pure pair of size <i>t</i> in a graph <i>G</i> is a pair <i>A</i>, <i>B</i> of disjoint subsets of <i>V</i>(<i>G</i>), each of cardinality at least <i>t</i>, such that <i>A</i> is either complete or anticomplete to <i>B</i>. It is known that, for every forest <i>H</i>, every graph on <span>\\(n\\ge 2\\)</span> vertices that does not contain <i>H</i> or its complement as an induced subgraph has a pure pair of size <span>\\(\\Omega (n)\\)</span>; furthermore, this only holds when <i>H</i> or its complement is a forest. In this paper, we look at pure pairs of size <span>\\(n^{1-c}\\)</span>, where <span>\\(0<c<1\\)</span>. Let <i>H</i> be a graph: does every graph on <span>\\(n\\ge 2\\)</span> vertices that does not contain <i>H</i> or its complement as an induced subgraph have a pure pair of size <span>\\(\\Omega (|G|^{1-c})\\)</span>? The answer is related to the <i>congestion</i> of <i>H</i>, the maximum of <span>\\(1-(|J|-1)/|E(J)|\\)</span> over all subgraphs <i>J</i> of <i>H</i> with an edge. (Congestion is nonnegative, and equals zero exactly when <i>H</i> is a forest.) Let <i>d</i> be the smaller of the congestions of <i>H</i> and <span>\\(\\overline{H}\\)</span>. We show that the answer to the question above is “yes” if <span>\\(d\\le c/(9+15c)\\)</span>, and “no” if <span>\\(d>c\\)</span>.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Combinatorica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00493-024-00117-z","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
A pure pair of size t in a graph G is a pair A, B of disjoint subsets of V(G), each of cardinality at least t, such that A is either complete or anticomplete to B. It is known that, for every forest H, every graph on \(n\ge 2\) vertices that does not contain H or its complement as an induced subgraph has a pure pair of size \(\Omega (n)\); furthermore, this only holds when H or its complement is a forest. In this paper, we look at pure pairs of size \(n^{1-c}\), where \(0<c<1\). Let H be a graph: does every graph on \(n\ge 2\) vertices that does not contain H or its complement as an induced subgraph have a pure pair of size \(\Omega (|G|^{1-c})\)? The answer is related to the congestion of H, the maximum of \(1-(|J|-1)/|E(J)|\) over all subgraphs J of H with an edge. (Congestion is nonnegative, and equals zero exactly when H is a forest.) Let d be the smaller of the congestions of H and \(\overline{H}\). We show that the answer to the question above is “yes” if \(d\le c/(9+15c)\), and “no” if \(d>c\).
图 G 中大小为 t 的纯对是 V(G) 的一对互不相交的子集 A、B,每个子集的卡片数至少为 t,使得 A 对 B 要么是完全的,要么是反完全的。众所周知,对于每个森林 H,每个不包含 H 或其补集作为诱导子图的 \(n\ge 2\) 个顶点上的图都有大小为 \(\Omega (n)\) 的纯对;此外,只有当 H 或其补集是一个森林时,这一点才成立。在本文中,我们关注的是大小为 \(n^{1-c}\) 的纯图对,其中 \(0<c<1\)。假设 H 是一个图:是否每一个不包含 H 或其补集作为诱导子图的顶点上的图都有大小为 \(\Omega (|G|^{1-c})\) 的纯对?答案与 H 的拥塞有关,即 H 的所有有边的子图 J 上的\(1-(|J|-1)/|E(J)|\)的最大值。(拥塞度是非负的,当 H 是森林时,拥塞度正好等于零。)设 d 是 H 的拥塞度和\(\overline{H}\)中较小的一个。我们证明,如果 \(d\le c/(9+15c)\) ,上述问题的答案是 "是";如果 \(d>c\) ,答案是 "否"。
期刊介绍:
COMBINATORICA publishes research papers in English in a variety of areas of combinatorics and the theory of computing, with particular emphasis on general techniques and unifying principles. Typical but not exclusive topics covered by COMBINATORICA are
- Combinatorial structures (graphs, hypergraphs, matroids, designs, permutation groups).
- Combinatorial optimization.
- Combinatorial aspects of geometry and number theory.
- Algorithms in combinatorics and related fields.
- Computational complexity theory.
- Randomization and explicit construction in combinatorics and algorithms.