A Sane Proof that COLk \le COL3

TL;DR

This paper presents a straightforward proof that for any k ≥ 3, the class of k-colorable graphs can be reduced to 3-colorable graphs, simplifying previous complex proofs based on the Cook-Levin theorem.

Contribution

It provides a simple, direct proof that COLk reduces to COL3 for all k ≥ 3, improving understanding of the relationships between graph coloring classes.

Findings

COLk reduces to COL3 for all k ≥ 3

Simplifies previous complex proofs using Cook-Levin theorem

Enhances understanding of graph coloring class reductions

Abstract

Let COLk be the set of all k-colorable graphs. It is easy to show that if a<b then COLa \le COLb (poly time reduction). Using the Cook-Levin theorem it is easy to show that if 3 \le a< b then COLb \le COLa. However this proof is insane in that it translates a graph to a formula and then the formula to a graph. We give a simple proof that COLk \le COL3.