### 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 |
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Pages (from-to) | 183-185 |

Number of pages | 3 |

Journal | Combinatorica |

Volume | 4 |

Issue number | 2-3 |

DOIs | |

Publication status | Published - Jun 1984 |

### Fingerprint

### Keywords

- AMS subject classification (1980): 05C15

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Mathematics(all)

### Cite this

*Combinatorica*,

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

**On coloring graphs with locally small chromatic number.** / Kierstead, H. A.; Szemerédi, E.; Trotter, W. T.

Research output: Contribution to journal › Article

*Combinatorica*, vol. 4, no. 2-3, pp. 183-185. https://doi.org/10.1007/BF02579219

}

TY - JOUR

T1 - On coloring graphs with locally small chromatic number

AU - Kierstead, H. A.

AU - Szemerédi, E.

AU - Trotter, W. T.

PY - 1984/6

Y1 - 1984/6

N2 - 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.

AB - 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.

KW - AMS subject classification (1980): 05C15

UR - http://www.scopus.com/inward/record.url?scp=51249185811&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=51249185811&partnerID=8YFLogxK

U2 - 10.1007/BF02579219

DO - 10.1007/BF02579219

M3 - Article

VL - 4

SP - 183

EP - 185

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 2-3

ER -