Caterpillars in Erdős–Hajnal
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Date
2019-05
Authors
Liebenau, Anita
Pilipczuk, Marcin
Seymour, Paul
Spirkl, Sophie
Advisor
Journal Title
Journal ISSN
Volume Title
Publisher
Elsevier
Abstract
Let T be a tree such that all its vertices of degree more than two lie on one path; that is, T is a caterpillar subdivision. We prove that there exists ε > 0 such that for every graph G with |V(G)| ≥ 2 not containing T as an induced subgraph, either some vertex has at least ε|V(G)| neighbours, or there are two disjoint sets of vertices A, B, both of cardinality at least ε|V(G)|, where there is no edge joining A and B.
A consequence is: for every caterpillar subdivision T, there exists c > 0 such that for every graph G containing neither of T and its complement as an induced subgraph, G has a clique or stable set with at least |V(G)| c vertices. This extends a theorem of Bousquet, Lagoutte and Thomassé [1], who proved the same when T is a path, and a recent theorem of Choromanski, Falik, Liebenau, Patel and Pilipczuk [2], who proved it when T is a “hook”.
Description
The final publication is available at Elsevier via https://doi.org/10.1016/j.jctb.2018.09.002. © 2019. 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
induced subgraphs, caterpillars, Erdős–Hajnal conjecture