Erdős-Hajnal for cap-free graphs

Maria Chudnovsky; Paul Seymour

doi:10.1016/j.jctb.2021.07.006

Erdős-Hajnal for cap-free graphs

Maria Chudnovsky, Paul Seymour

Mathematics

Princeton University

Research output: Contribution to journal › Article › peer-review

Abstract

A “cap” in a graph G is an induced subgraph of G that consists of a cycle of length at least four, together with one further vertex that has exactly two neighbours in the cycle, adjacent to each other, and the “house” is the smallest, on five vertices. It is not known whether there exists ε>0 such that every graph G containing no house has a clique or stable set of cardinality at least |G|^ε; this is the smallest open case of the Erdős-Hajnal conjecture and has been the subject of much study. We prove that there exists ε>0 such that every graph G with no cap has a clique or stable set of cardinality at least |G|^ε.

Original language	American English
Pages (from-to)	417-434
Number of pages	18
Journal	Journal of Combinatorial Theory. Series B
Volume	151
DOIs	https://doi.org/10.1016/j.jctb.2021.07.006
State	Published - Nov 2021

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics
Computational Theory and Mathematics

Keywords

Cap
Erdos-Hajnal conjecture
Hole with a hat
Induced subgrpah
Non-crossing separations

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10.1016/j.jctb.2021.07.006

Cite this

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title = "Erd{\H o}s-Hajnal for cap-free graphs",

abstract = "A “cap” in a graph G is an induced subgraph of G that consists of a cycle of length at least four, together with one further vertex that has exactly two neighbours in the cycle, adjacent to each other, and the “house” is the smallest, on five vertices. It is not known whether there exists ε>0 such that every graph G containing no house has a clique or stable set of cardinality at least |G|ε; this is the smallest open case of the Erd{\H o}s-Hajnal conjecture and has been the subject of much study. We prove that there exists ε>0 such that every graph G with no cap has a clique or stable set of cardinality at least |G|ε.",

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AU - Chudnovsky, Maria

AU - Seymour, Paul

PY - 2021/11

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N2 - A “cap” in a graph G is an induced subgraph of G that consists of a cycle of length at least four, together with one further vertex that has exactly two neighbours in the cycle, adjacent to each other, and the “house” is the smallest, on five vertices. It is not known whether there exists ε>0 such that every graph G containing no house has a clique or stable set of cardinality at least |G|ε; this is the smallest open case of the Erdős-Hajnal conjecture and has been the subject of much study. We prove that there exists ε>0 such that every graph G with no cap has a clique or stable set of cardinality at least |G|ε.

AB - A “cap” in a graph G is an induced subgraph of G that consists of a cycle of length at least four, together with one further vertex that has exactly two neighbours in the cycle, adjacent to each other, and the “house” is the smallest, on five vertices. It is not known whether there exists ε>0 such that every graph G containing no house has a clique or stable set of cardinality at least |G|ε; this is the smallest open case of the Erdős-Hajnal conjecture and has been the subject of much study. We prove that there exists ε>0 such that every graph G with no cap has a clique or stable set of cardinality at least |G|ε.

KW - Cap

KW - Erdos-Hajnal conjecture

KW - Hole with a hat

KW - Induced subgrpah

KW - Non-crossing separations

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DO - 10.1016/j.jctb.2021.07.006

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JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

ER -

Erdős-Hajnal for cap-free graphs

Abstract

ASJC Scopus subject areas

Keywords

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