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 …
