Planting trees on 3-spheres
We look at the problem to find the possible arboricities which a 3-sphere can have.
Tag: graph theory
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 …
Product Formula For Curvature
The Curvature of graphs multplies under the Shannon product (strong product).
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 …
More theorems about graphs
Last week, I practiced a bit more enhanced talk presentation style in which, rather than with slides, the content is spoken and then enhanced in the video using additonal illustrations. The presentation deals with some things I have done in graph theory which I consider as part of quantum calculus …
10 theorems on discrete manifolds
10 theorems about discrete manifolds were featured in a youtube video.
Homotopy Manifolds
The problem of discretization It is a question which probably was pondered first by philosophers like Democrit or Archimedes. What is the nature of space? It is made of discrete stuff or is it a true continuum? As Plato already noted, such questions border to being pointless as we live …
The Hamiltonian Manifold Theorem
We prove that connected combinatorial manifolds of positive dimension define finite simple graphs which are Hamiltonian.