### Abstract

In 1973, P. Erdös conjectured that for each kε2, there exists a constant c_{ k} so that if G is a graph on n vertices and G has no odd cycle with length less than c_{ k} n^{ 1/k}, then the chromatic number of G is at most k+1. Constructions due to Lovász and Schriver show that c_{ k}, if it exists, must be at least 1. In this paper we settle Erdös' conjecture in the affirmative. We actually prove a stronger result which provides an upper bound on the chromatic number of a graph in which we have a bound on the chromatic number of subgraphs with small diameter.

Original language | English (US) |
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Pages (from-to) | 183-185 |

Number of pages | 3 |

Journal | Combinatorica |

Volume | 4 |

Issue number | 2-3 |

DOIs | |

State | Published - Jun 1 1984 |

Externally published | Yes |

### Keywords

- AMS subject classification (1980): 05C15

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics

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## Cite this

Kierstead, H. A., Szemerédi, E., & Trotter, W. T. (1984). On coloring graphs with locally small chromatic number.

*Combinatorica*,*4*(2-3), 183-185. https://doi.org/10.1007/BF02579219