It is well known that you can color a graph G of maximum degree d greedily with d+1 colors. Moreover, this bound is tight, since it is reached by the cliques. Johansson proved with a pseudo-random coloring scheme that you can color triangle-free graphs of maximum degree d with no more than O(d/log d) colors. This result has been recently improved to (1+ε)(d/log d) for any ε>0 when d is big enough. This is tight up to a multiplicative constant, since you can pseudo-randomly construct a family of graphs of maximum degree d, arbitrary large girth, and chromatic number bigger than d / (2 log d). Although these are really nice results, they are only true for big degrees, and there remains a lot to say for small degree graphs. When the graphs are of small degree, it is interesting to consider the fractional chromatic number instead, since it has infinitely many possible values – note that cubic graphs are either bipartite, the clique K4, or of chromatic number 3. It has already been settled that the maximum fractional chromatic number over the triangle-free cubic graphs is 14/5. I will present a systematic method to compute upper bounds for the independence ratio of graphs of given (small!) degree and girth, which can sometimes lead to upper bounds for the fractional chromatic number, and can be adapted to any family of small degree graphs under some local constraints.