Caterpillars in Erdős–Hajnal

Anita Liebenau, Marcin Pilipczuk, Paul Douglas Seymour, Sophie Spirkl

Research output: Contribution to journalArticle

3 Scopus citations

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”.

Original languageEnglish (US)
JournalJournal of Combinatorial Theory. Series B
DOIs
StateAccepted/In press - Jan 1 2018

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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