The Three Tree Theorem

TL;DR

This paper proves that the arboricity of a discrete 2-sphere is 3, showing that such graphs can be vertex 4-colored with all Kempe chains as forests, and extends this to planar graphs.

Contribution

It establishes the three tree theorem for 2-sphere graphs, demonstrating their arboricity is 3 and that the trees can intersect all triangles, advancing understanding of graph coloring and arboricity.

Findings

Every 2-sphere graph (not a prism) is vertex 4-colorable with Kempe chains as forests.

The arboricity of a discrete 2-sphere is exactly 3.

Planar graphs have arboricity at most 3, based on this work.

Abstract

We prove that every 2-sphere graph different from a prism can be vertex 4-colored in such a way that all Kempe chains are forests. This implies the following three tree theorem: the arboricity of a discrete 2-sphere is 3. Moreover, the three trees can be chosen so that each hits every triangle. A consequence is a result of an exercise in the book of Bondy and Murty based on work of A. Frank, A. Gyarfas and C. Nash-Williams: the arboricity of a planar graph is less or equal than 3.