Complex Analysis and Existence Problems for plane Graphs

2026/07/27 by Fedor Pakovich
#math.CO #math.AG #math.CV #math.GT

paper · pdf

Abstract

We show that a variety of known and new results concerning connected plane graphs whose vertex and face degrees satisfy prescribed uniformity conditions with at most two exceptions can be deduced from recent results on the Hurwitz existence problem regarding the realizability of branch patterns of rational functions. Our method also yields a description of the Belyi functions corresponding to such graphs.

Citations

• Fullerenes and Belyi functions
• 2-nearly Platonic graphs are unique
• A note on nearly Platonic graphs with connectivity one
• On the postcritical set of a rational map
• On generalized Lattès maps
• On rational functions whose normalization has genus zero or one
• A note on nearly platonic graphs
• Minimum Degree of the Difference of Two Polynomials over Q, and Weighted Plane Trees
• Fullerene-like spheres with faces of negative curvature
• Solution of the Hurwitz problem for Laurent polynomials
• On the existence of branched coverings between surfaces with prescribed branch data, II

Related

• Integral and rational graphs in the plane
• Irreducible components of sets of points in the plane that satisfy distance conditions
• Uniform Spanning Forests of Planar Graphs
• Every planar graph with the Liouville property is amenable
• Graphs that are not complete pluripolar