No abstract

The Lovász theta number of a graph G can be viewed as a semidefinite programming relaxation of the stability number of G. It has recently been shown that a copositive strengthening of this semidefinite program in fact equals the stability number of G. We introduce a related strengthening of the Lovász theta number toward the chromatic number of G, which is shown to be equal to the fractional chromatic number of G. Solving copositive programs is NP-hard. This motivates the study of tractable approximations of the copositive cone. We investigate the Parrilo hierarchy to approximate this cone and provide computational simplifications for the approximation of the chromatic number of vertex transitive graphs. We provide some computational results indicating that the Lovász theta number can be strengthened significantly toward the fractional chromatic number of G on some Hamming graphs.

The Lovász -function is a well-known polynomial lower bound on the chromatic number. Any near-optimal solution of its semidefinite programming formulation carries valuable information on how to color the graph. A self-contained presentation of the role of this formulation in obtaining heuristics for the graph coloring problem is presented. These heuristics could be useful for coloring medium sized graphs as numerical results on DIMACS benchmark graphs indicate.