A curvature notion for planar graphs stable under planar duality

Woess [30] introduced a curvature notion on the set of edges of a planar graph, called Ψ-curvature in our paper, which is stable under the planar duality. We study geometric and combinatorial properties for the class of infinite planar graphs with non-negative Ψ-curvature. By using the discharging method, we prove that for such an infinite graph the number of vertices (resp. faces) of degree k, except k=3,4 or 6, is finite. As a main result, we prove that for an infinite planar graph with non-negative Ψ-curvature the sum of the number of vertices of degree at least 8 and the number of faces of degree at least 8 is at most one.