Calculus without limits ## 3 trees and 4 colors

The three tree theorem follows from an upgrade of the 4 color theorem.

## The Three Tree Theorem

The three tree theorem tells that any discrete 2-sphere has arboricity exctly 3.

## Arboricity, Dimension, Category

We aim to show that the Lusternik-Schnirelmann category of a graph is bound by the augmented dimension. We can try to prove this by using tree coverings and so look at arboricity.

## Incidence and Intersection

Barycentric and Connection graphs Barycentric graphs depend on incidence, connection graphs on intersection. Here are some examples from this blog. Both graphs have as the vertex set the complete subgraphs of the graph. In the connection graph, we take the intersection, in the Barycentric case, we take incidence. Here are …

## More on Graph Arithmetic

A few more remarks [PDF] about graph arithmetic. Now on the ArXiv. (Previous documents are here (June 2017 ArXiv), here (August 2017 ArXiv) and here (May 2019 ArXiv). The talk below on youtube was used for me to get organized a bit. It does not look like much has changed, …

## Topology of Manifold Coloring

Last summer I have had some fun with codimension 2 manifolds M in a purely differential geometric setting: a positive curvature d-manifold which admits a circular action of isometries has a fixed point set K which consists of even codimension positive curvature manifold. The Grove-Searle situation https://arxiv.org/abs/2006.11973 is when K …