A Unified Spiral Chain Coloring Algorithm for Planar Graphs

TL;DR

This paper introduces a unified spiral chain coloring algorithm for planar graphs, providing proofs for several key conjectures including the four color theorem, Vizing's total coloring, and Groetzsch's theorem, demonstrating its broad applicability.

Contribution

It presents a novel unified spiral chain coloring approach that simplifies proofs of major planar graph coloring conjectures and theorems.

Findings

Proof of the four color theorem using spiral chain coloring

Validation of Vizing's total coloring conjecture for maximum degree six

Short proof of Groetzsch's theorem on 3-colorability of triangle-free planar graphs

Abstract

In this paper we have given a unified graph coloring algorithm for planar graphs. The problems that have been considered in this context respectively, are vertex, edge, total and entire colorings of the planar graphs. The main tool in the coloring algorithm is the use of spiral chain which has been used in the non-computer proof of the four color theorem in 2004. A more precies explanation of the proof of the four color theorem by spiral chain coloring is also given in this paper. Then we continue to spiral-chain coloring solutions by giving the proof of other famous conjectures of Vizing's total coloring and planar graph conjectures of maximum vertex degree six. We have also given the proof of a conjecture of Kronk and Mitchem that any plane graph of maximum degree "Delta" is entirely ("Delta"+4)-colorable.The last part of the paper deals with the three colorability of planar graphs under the spiral chain coloring. We have given an efficient and short proof of the Groetzsch's Theorem that triangle-free planar graphs are 3-colorable.