### Abstract

Combining recent results on colorings and Ramsey theory, we show that if G is a triangle-free graph with e edges then the chromatic number of G is at most cel(1/3)(log e)(-2/3) for some constant c. In a previous paper, we found an upper bound on the chromatic number of a triangle-free graph of genus g. Using the new result, we slightly improve this bound to cg(1/3)(log g)(-2/3). Both bounds are best possible, up to a constant multiple. (C) 2000 Elsevier Science B.V. All rights reserved.

Original language | English |
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Journal | Discrete Mathematics |

Volume | 219 |

Issue number | 1-3 |

Pages (from-to) | 275-277 |

ISSN | 0012-365X |

DOIs | |

Publication status | Published - 2000 |

## Cite this

Thomassen, C., & Gimbel, J. (2000). Coloring triangle-free graphs with fixed size.

*Discrete Mathematics*,*219*(1-3), 275-277. https://doi.org/10.1016/S0012-365X(00)00087-X