A generic NP-hardness proof for a variant of Graph Coloring

Abstract

In this note, a direct proof is given of the NP-completeness of a variant of GRAPH COLORING, i.e., a generic proof is given, similar to the proof of Cook of the NP-completeness of SATISFIABILITY. Then, transformations from this variant of GRAPH COLORING to INDEPENDENT SET and to SATISFIABILITY are given. These proofs could be useful in an educational setting, where basics of the theory of NP-completeness must be explained to students whose background in combinatorial optimisation and/or graph theory is stronger than their background in logic. In addition, I believe that the proof given here is slightly easier than older generic proofs of NP-completeness.