AN UPPER BOUND ON THE FRACTIONAL CHROMATIC NUMBER OF TRIANGLE-FREE SUBCUBIC GRAPHS

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 chi(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 chi(f)(G) <= 3 - 3/64 approximate to 2.953. Lu and Peng improved the bound to chi(f) (G) <= 3 - 3/43 approximate to 2.930. Recently, Ferguson, Kaiser, and Kral' proved that chi(f) (G) <= 32/11 approximate to 2.909. In this paper, we prove that chi(f) (G) <= 43/15 approximate to 2.867.