Uma conjectura de Erdos e Hajnal

Abstract

A result by Erdős shows that there exist graphs with arbitrarily large chromatic number and girth. Thus a sufficiently large clique contains a subgraph with large chromatic number and girth, but many graphs do not have a large clique, hence it is interesting to find a different condition that guarantees the existence of a subgraph with large chromatic number and girth. A conjecture by Erdős and Hajnal states that every graph with sufficiently large chromatic number contains a subgraph with large chromatic number and girth. The objective of this text is to study this conjecture.The text begins with a brief discussion of triangle-free constructions. In particular, we show a construction by Codenotti, Pudlák and Resta, based on projective planes.The main topic begins with a proof by Rödl, that every graph with sufficiently large chromatic number contains a triangle-free subgraph with large chromatic number. We follow with a proof that every graph with sufficiently large chromatic number contains a large odd cycle.We then show a result by Mohar and Wu, which shows that the Kneser graphs respect the Erdős-Hajnal conjecture. Another result by Gábor Tardos proves that shift graphs also respect the Erdős-Hajnal conjecture. Finally, we show some brief contributions about type graphs, showing some cases that follow the Erdős-Hajnal conjecture.