A polynomial 3-colorability algorithm with automatic generation of NO   3-colorability (i.e. Co-NP) short proofs
  In this paper, an algorithm for determining 3-colorability, i.e. the decision problem (YES/NO), in planar graphs is presented. The algorithm, although not exact (it could produce false positives) has two very important features: (i) it has polynomial complexity and (ii) for every "NO" answer, a "short" proof is generated, which is of much interest since 3-colorability is a NP-complete problem and thus its complementary problem is in Co-NP. Hence the algorithm is exact when it determines that a given planar graph is not 3-colorable since this is verifiable via an automatic generation of short formal proofs (also human-readable). 
Jose Antonio Martin H
cs.DM cs.CC cs.DS math.CO