Pure Pairs.VIII.排除稀疏图

IF 1 2区 数学 Q1 MATHEMATICS
Alex Scott, Paul Seymour, Sophie Spirkl
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引用次数: 0

摘要

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"。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

Pure Pairs. VIII. Excluding a Sparse Graph

Pure Pairs. VIII. Excluding a Sparse Graph

A pure pair of size t in a graph G is a pair AB 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 \(d>c\).

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来源期刊
Combinatorica
Combinatorica 数学-数学
CiteScore
1.90
自引率
0.00%
发文量
45
审稿时长
>12 weeks
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.
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