A curvature notion for planar graphs stable under planar duality

2019/09/17 by Akama, Yohji, Hua, Bobo, Su, Yanhui +1
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1909.07825

Abstract

Woess \citeWoess98 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.

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