Pure pairs. I. Trees and linear anticomplete pairs
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Date
2020-12-02
Authors
Chudnovsky, Maria
Scott, Alex
Seymour, Paul
Spirkl, Sophie
Journal Title
Journal ISSN
Volume Title
Publisher
Elsevier
Abstract
The Erdős-Hajnal conjecture asserts that for every graph H there is a constant c > 0 such that every
graph G that does not contain H as an induced subgraph has a clique or stable set of cardinality at
least |G|c. In this paper, we prove a conjecture of Liebenau and Pilipczuk [11], that for every forest
H there exists c > 0, such that every graph G with |G| > 1 contains either an induced copy of H, or
a vertex of degree at least c|Gj|, or two disjoint sets of at least c|G| vertices with no edges between
them. It follows that for every forest H there exists c > 0 such that, if G contains neither H nor its
complement as an induced subgraph, then there is a clique or stable set of cardinality at least |G|c.
Description
The final publication is available at Elsevier via https://doi.org/10.1016/j.aim.2020.107396. © 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Keywords
Erdős-Hajnal conjecture, induced subgraphs, forests