By Bonnet-Meyer’s theorem, if a manifold has positive curvature everywhere, then it is compact.

For discrete analogue of this theorem, we will consider tessellations (i.e., graphs embedded on surfaces) for manifolds, and combinatorial curvature or Forman curvature for the curvature of manifolds. This kind of research was initiated by Higuchi, and studied by DeVos-Mohar, Chen-Chen and so on.

We wish to classify and enumerate tessellations with these discrete curvatures nonnegative everywhere.

As partial results for this research direction, we discuss

(1) regular spherical/hyperbolic polyhedral surfaces and the unit sphere by using combinatorial curvature (joint work with Bobo HUA, Yanhui Su and Lili Wang), and

(2) the enumeration of planar graphs with zero Forman curvature everywhere. 

For (2), we explain the usefulness of Brinkmann-MacKay’s canonical construction path technique to enumerate algorithmically all the spherical quadrangulations/ pentagulations.

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