Toward a language theoretic proof of the four color theorem

TL;DR

This paper explores a language theoretic approach to the four color theorem by analyzing common parse words for pairs of binary trees, linking tree parsing problems to graph coloring.

Contribution

It introduces a novel connection between tree parsing and graph coloring, providing enumeration methods and reduction strategies relevant to a language theoretic proof of the four color theorem.

Findings

Enumerated common parse words for specific tree families

Developed reduction techniques for larger tree pairs

Linked tree parsing problems to planar graph colorability

Abstract

This paper considers the problem of showing that every pair of binary trees with the same number of leaves parses a common word under a certain simple grammar. We enumerate the common parse words for several infinite families of tree pairs and discuss several ways to reduce the problem of finding a parse word for a pair of trees to that for a smaller pair. The statement that every pair of trees has a common parse word is equivalent to the statement that every planar graph is four-colorable, so the results are a step toward a language theoretic proof of the four color theorem.