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 χf (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 χf (G) ≤ 3 - 3/64 ≈ 2. 953. Lu and Peng improved the bound to χf (G) ≤ 3 - 3/43 ≈ 2.930. Recently, Ferguson, Kaiser, and Král' proved that χf (G) ≤ 32/11 ≈ 2.909. In this paper, we prove that χf (G) ≤ 43/15 ≈ 2.867.
All Science Journal Classification (ASJC) codes
- Fractional chromatic number
- Subcubic graphs
- Triangle-free graphs