### Abstract

An (a : b)-coloring of a graph G is a function f which maps the vertices of G into b-element subsets of some set of size a in such a way that f (u) is disjoint from f (v) for every two adjacent vertices u and v in G. The fractional chromatic number χ_{f} (G) is the infimum of a/b over all pairs of positive integers a, b such that G has an (a : b)-coloring. Heckman and Thomas conjectured that the fractional chromatic number of every triangle-free graph G of maximum degree at most three is at most 2.8. Hatami and Zhu proved that χ_{f} (G) ≤ 3 - 3/64 ≈ 2. 953. Lu and Peng improved the bound to χ_{f} (G) ≤ 3 - 3/43 ≈ 2.930. Recently, Ferguson, Kaiser, and Král' proved that χ_{f} (G) ≤ 32/11 ≈ 2.909. In this paper, we prove that χ_{f} (G) ≤ 43/15 ≈ 2.867.

Original language | English (US) |
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Pages (from-to) | 1102-1136 |

Number of pages | 35 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 28 |

Issue number | 3 |

DOIs | |

State | Published - Jan 1 2014 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Keywords

- Fractional chromatic number
- Subcubic graphs
- Triangle-free graphs

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## Cite this

*SIAM Journal on Discrete Mathematics*,

*28*(3), 1102-1136. https://doi.org/10.1137/120900678