Graphs on surfaces with positive Forman curvature or corner curvature

On one hand, we study the class of graphs on surfaces, satisfying tessellation properties, with positive Forman curvature on each edge. Via medial graphs, we provide a new proof for the finiteness of the class, and give a complete classification. On the other hand, we classify the class of graphs on surfaces with positive corner curvature.
Comment: 25 pages, 9 figures, 12 tables