Mathematical Programming

, Volume 109, Issue 2–3, pp 345–365 | Cite as

Semidefinite programming relaxations for graph coloring and maximal clique problems



The semidefinite programming formulation of the Lovász theta number does not only give one of the best polynomial simultaneous bounds on the chromatic number χ(G) or the clique number ω(G) of a graph, but also leads to heuristics for graph coloring and extracting large cliques. This semidefinite programming formulation can be tightened toward either χ(G) or ω(G) 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.


Lovász theta number Chromatic number Clique number Cutting planes 


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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Ekonomsko-poslovna fakultetaUniverza v MariboruMariborSlovenia
  2. 2.Institut für MathematikUniversität KlagenfurtKlagenfurtAustria

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