The Four Color Theorem and Trees

TL;DR

This paper explores the connection between the Four Color Theorem and tree operations, proposing a conjecture about special colorings of maps represented by trees that preserve proper coloring through transformations.

Contribution

It introduces a new conjecture linking map colorings with tree transformations and proves it for certain classes of trees, expanding understanding of the Four Color Theorem.

Findings

Conjecture relates map coloring to tree transformations.

Proof of the conjecture for specific classes of trees.

Links between the Four Color Theorem and associative law in tree operations.

Abstract

Connection of the Four Color Theorem (FCT) with some operations on trees is described. L.H. Kauffman's theorem about FCT and vector cross product is discussed. Operation of transplantation on trees linked with the move of brackets according to the associative law is used to formulate a conjecture. When map is represented as a tying of the trees this conjecture proposes the existence of special coloring of this map. This coloring makes possible successive transplantations such that one of these trees is transformed to another, and all the intermediate maps are colored properly. It means (in terms of L.H. Kauffman) that not only for two different positions of brackets in a product of n factors the associative law works (on special values of factors) but also there is a way of moving brackets that for all intermediate positioning of brackets the associative law is obeyed. Some classes of trees for which the conjecture is proved are presented.