FRACTIONAL CHROMATIC NUMBER, MAXIMUM DEGREE, AND GIRTH

We introduce a new method for computing bounds on the independence number and fractional chromatic number of classes of graphs with local constraints and apply this method in various scenarios. We establish a formula that generates a general upper bound for the fractional chromatic number of triangle-free graphs of maximum degree Delta >= 3. This upper bound matches that deduced from the fractional version of Reed's bound for small values of Delta , and improves it when Delta >= 17, transitioning smoothly to the best possible asymptotic regime, barring a breakthrough in Ramsey theory. Focusing on smaller values of Delta , we also demonstrate that every graph of girth at least 7 and maximum degree Delta has fractional chromatic number at most 1 + min(k epsilon N) 2 Delta+2(k-3)/k . In particular, the fractional chromatic number of a graph of girth 7 and maximum degree Delta is at most 2 Delta+9/5 when Delta epsilon [3, 8], at most Delta+7/3 when Delta epsilon [8, 20], at most 2 Delta+23/7 when Delta epsilon [20, 48], and at most Delta/1 + 5 when Delta epsilon [48, 112]. In addition, we also obtain new lower bounds on the independence ratio of graphs of maximum degree Delta epsilon {3, 4, 5} and girth g epsilon {6, . . . ,12}, notably 1/3 when (Delta , g) = (4,10) and 2/7 when (Delta , g) = (5, 8).