# The Erdös--Hajnal Conjecture for Long Holes and Antiholes

2 GOAL - Graphes, AlgOrithmes et AppLications
LIRIS - Laboratoire d'InfoRmatique en Image et Systèmes d'information
3 MC2 - Modèles de calcul, Complexité, Combinatoire
LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : Erdös and Hajnal conjectured that for every graph $H$, there exists a constant $c_H$ such that every graph $G$ on $n$ vertices which does not contain an induced copy of $H$ has a clique or a stable set of size $n^{c_H}$. We prove that for every $k$ there exists $c_k>0$ such that every graph $G$ on $n$ vertices not inducing a cycle of length at least $k$ nor its complement contains a clique or a stable set of size at least $n^{c_k}$.
Document type :
Journal articles
Domain :

https://hal-lirmm.ccsd.cnrs.fr/lirmm-01347304
Contributor : Isabelle Gouat <>
Submitted on : Wednesday, July 20, 2016 - 5:22:02 PM
Last modification on : Friday, April 19, 2019 - 11:12:03 AM

### Citation

Marthe Bonamy, Nicolas Bousquet, Stéphan Thomassé. The Erdös--Hajnal Conjecture for Long Holes and Antiholes. Siam Journal on Discrete Mathematics, Society for Industrial and Applied Mathematics, 2016, 30 (2), pp.1159-1164. ⟨10.1137/140981745⟩. ⟨lirmm-01347304⟩

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