The Three Tree Theorem
Combinatorics
2023-09-06 v1 Discrete Mathematics
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.
Keywords
Cite
@article{arxiv.2309.01869,
title = {The Three Tree Theorem},
author = {Oliver Knill},
journal= {arXiv preprint arXiv:2309.01869},
year = {2023}
}
Comments
17 pages, 3 figures