Simple Proofs of the Strong Perfect Graph Theorem Using Polyhedral Approaches and Proving P=NP as a Conclusion

Abstract

The strong perfect graph theorem is the proof of the famous Berge’s conjecture that the graph is perfect if and only if it is free of odd holes and odd anti-holes. The conjecture was settled after 40 years in 2002 by Maria Chudnovsky et. al. and the proof was published in 2006. However, that proof is lengthy and intricate and using a combinatorial approach. We provide simple short proofs of the strong perfect graph theorem using polyhedral methods. We first prove the weak perfect graph theorem by polyhedral methods and use that to prove the strong perfect graph theorem. Our proofs emerge naturally from our work to calculate the capacity of multihop wireless networks. As a corollary of our proofs techniques we prove P=NP by proving there is an algorithm to find the maximum independent set or the independence number of any graph in polynomial time as a function of the number of the graph vertices.