On Equivalent Color Transform and Four Coloring Theorem

TL;DR

This paper introduces an equivalent color transform (ECT) method for graph coloring and presents a simple proof of the four color theorem for planar graphs using this approach.

Contribution

It proposes a novel ECT technique for minimal graph coloring and offers a new, simplified proof of the four color theorem for planar graphs.

Findings

ECT contracts color classes to single vertices

Produces complete graphs K_k from minimal colorings

Provides a simple proof for the four color theorem

Abstract

In this paper, we apply an equivalent color transform (ECT) for a minimal $k$-coloring of any graph $G$. It contracts each color class of the graph to a single vertex and produces a complete graph $K_k$ for $G$ by removing redundant edges between any two vertices. Based on ECT, a simple proof for four color theorem for planar graph is then proposed.