[Update: May 28: I have an expository paper on this uploaded now. Ha a few days for focused writing.] I encountered the Moebius Kantor Graph when trying to get better geodesic sheets in discrete 3-manifolds. For , the smallest 3-sphere, all dual spheres of edges are graphs and the geodesic …
A good discrete differential geometry should mirror the continuum as closely as possible. Positive curvature manifolds should be bounded, the list of positive curvature manifolds should be identical to the ones we know in the continuum. Last year we had looked at geodesic patches and defined sectional curvature by defining …
We have seen last time that describing many particles moving on a manifold is best done by keeping on each cell a tag telling how many particles there are there. In other words, we do not evolve a sequence of maps but a divisor . This is more effective if …
Curvature is a local notion in a finite geometry that adds up to a topological invariant. This is Gauss-Bonnet. I’m only really interested in curvatures that satisfy this exactly. This does not exclude sectional curvature, the latest venture because sectional curvature integrated over a geodesic sheet is the Euler characteristic …
Here are two fundamental problems for discrete q-manifolds (a notion which can be defined for the trinity of geometric structures, that is for finite simple graphs as well as for simplicial complexes or delta sets). As for 1, one could ask that every embedded wheel graph has positive curvature. This …
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 …
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 …
