The fractional chromatic number χF(G) of a graph G is a well-known lower bound for its chromatic number χ(G). Experiments reported in the literature show that using χF(G), instead of the size of the maximum clique, can be much more efficient to drive a search in a branch-and-bound algorithm for finding χ(G). One difficulty though is to deal with the linear program that is known for χF(G). Such a formulation has an exponential number of variables and demands an expensive column generation process. In this work, we investigate the use of an alternative formulation to find a lower bound for χF(G), which has a linear number of variables but an exponential number of constraints. We use a cutting plane method to deal with this inconvenience. Some separation heuristics are proposed, and some computational experiments were carried out. They show that values very close to χF(G) (in many cases exact values) are found in less time than that required by the column generation.
chromatic number; cutting planes; combinatorial optimization
