Herbert Groetzsch and Jan Mycielski worked in the 50ies on the chromatology of triangle free graphs. Groetzsch’s 3 color theorem assures that planar triangle free graphs have chromatic number 3 or less. Mycielski defined an operation on graphs which preserves triangle free graphs and increases the chromatic number exactly by …
Coloring manifolds is a wonderful theme because it is not that well studied and so has many low hanging fruits. Here is an other one. For d-manifolds, the chromatic number is between d+1 and 2d+2 and a growth rate (3d+1)/2 is observed and reasonably conjectured upper bound. I started to …
The Regge approach to discrete relativity is also related to graph coloring problems of d-manifolds. While one can look at the length of the dual sphere of a codimension-2 simplex as a notion of curvature, the distinction of whether this circle has even or odd length is relevant when wanting …
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 …
10 theorems about discrete manifolds were featured in a youtube video.
Dehn-Sommerville relations are a symmetry for a class of geometries which are of Euclidean nature.
