The semidefinite programming formulation of the Lovasz theta number does not only give one of the best polynomial simultaneous bounds on the chromatic number and the clique number of a graph, but also leads to heuristics for graph coloring and extracting large cliques. This semidefinite programming formulation can be tightened toward either number by adding several types of cutting planes. We explore several such strengthenings, and show that some of them can be computed with the same effort as the theta number. We also investigate computational simplifications for graphs with rich automorphism groups.
To appear in Mathematical Programming B.