Pure pairs. IX. Transversal trees

TL;DR

This paper investigates the existence and size of pure pairs in graphs with vertex partitions, focusing on how the absence of certain transversal subgraphs influences pure pair properties.

Contribution

It presents new results and open questions about the size of pure pairs in graphs excluding specific transversal subgraphs, advancing understanding of graph structure.

Findings

Identifies conditions under which large pure pairs must exist

Provides bounds on pure pair sizes relative to excluded transversal subgraphs

Highlights open problems for future research in graph theory

Abstract

Fix k>0, and let G be a graph, with vertex set partitioned into k subsets (`blocks') of approximately equal size. An induced subgraph of G is transversal (with respect to this partition) if it has exactly one vertex in each block (and therefore it has exactly k vertices). A pure pair in G is a pair X,Y of disjoint subsets of V(G) such that either all edges between X,Y are present or none are; and in the present context we are interested in pure pairs (X,Y) where each of X,Y is a subset of one of the blocks, and not the same block. This paper collects several results and open questions concerning how large a pure pair must be present if various types of transversal subgraphs are excluded.