Pure pairs. VIII. Excluding a sparse graph

TL;DR

This paper investigates the existence of large pure pairs in graphs excluding a certain induced subgraph or its complement, revealing a relationship between the size of pure pairs and the congestion of the excluded subgraph.

Contribution

It establishes conditions based on congestion for the existence of large pure pairs in graphs avoiding a fixed induced subgraph or its complement.

Findings

Pure pairs of size proportional to $n^{1-c}$ exist under certain congestion conditions.

The existence of large pure pairs depends on the congestion measure of the excluded subgraph.

The paper characterizes when such pure pairs are guaranteed based on the congestion of $H$ and its complement.

Abstract

A pure pair of size $t$ in a graph $G$ is a pair $A,B$ of disjoint sets of $t$ vertices such that $A$ is either complete or anticomplete to $B$. It is known that, for every forest $H$, every graph on $n\ge2$ 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 $n\ge2$ vertices that does not contain $H$ or its complement as an induced subgraph have a pure pair $A,B$ with $|A|,|B|\ge \Omega(|G|^{1-c})$,? 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 $d>c$.