Gr\"obner Basis Convex Polytopes and Planar Graph

TL;DR

This paper demonstrates that the Gr"obner basis of certain polynomial ideals can be used to prove that all planar graphs are 4-colorable, leveraging algebraic and combinatorial properties.

Contribution

It introduces a novel algebraic approach using Gr"obner bases to establish the 4-colorability of planar graphs, connecting algebraic geometry with graph theory.

Findings

Proves all planar graphs are 4-colorable using algebraic methods.

Utilizes Gr"obner basis of polynomial ideals in graph coloring.

Shows the exclusion of K5 as a non-planar graph supports the proof.

Abstract

Using the Gr\"obner basis of an ideal generated by a family of polynomials we prove that every planar graph is 4-colorable. Here we also use the fact that the complete graph of 5 vertices is not included in any planar graph.