On the Three Colorability of Planar Graphs

TL;DR

This paper introduces a new criterion for three colorability of planar graphs, generalizing existing theorems by linking cycle properties and parity symmetry in triangulated graphs.

Contribution

It presents a novel three colorability criterion for planar graphs based on cycle parity symmetry, extending Heawood and Grotszch theorems.

Findings

Triangulated planar graphs with disjoint holes are 3-colorable if each hole satisfies parity symmetry.

The criterion generalizes previous theorems on cycle lengths and colorability.

Provides a new perspective on cycle structures influencing graph colorability.

Abstract

The chromatic number of an planar graph is not greater than four and this is known by the famous four color theorem and is equal to two when the planar graph is bipartite. When the planar graph is even-triangulated or all cycles are greater than three we know by the Heawood and the Grotszch theorems that the chromatic number is three. There are many conjectures and partial results on three colorability of planar graphs when the graph has specific cycles lengths or cycles with three edges (triangles) have special distance distributions. In this paper we have given a new three colorability criteria for planar graphs that can be considered as an generalization of the Heawood and the Grotszch theorems with respect to the triangulation and cycles of length greater than 3. We have shown that an triangulated planar graph with disjoint holes is 3-colorable if and only if every hole satisfies the parity symmetric property, where a hole is a cycle (face boundary) of length greater than 3.