Exact Solution of the Three-color Problem on a Random Lattice

TL;DR

This paper provides an exact solution to Baxter's three-color problem on a random planar graph using a random-matrix approach, revealing the approximate coloring density per vertex.

Contribution

It introduces an exact analytical solution for the three-color problem on random lattices via a matrix model, a novel approach in this context.

Findings

Number of three-colorings per vertex is approximately 0.9843.

Uses random-matrix formulation to solve a combinatorial problem.

Provides insights into coloring properties of random planar graphs.

Abstract

We present the exact solution of the Baxter's three-color problem on a random planar graph, using the random-matrix formulation of the problem, given by B. Eynard and C. Kristjansen. We find that the number of three-coloring of an infinite random graph is 0.9843 per vertex.