Color or cover

TL;DR

This paper proves a theorem about the adjacency of exceptional vertices in triangulated spheres with degree conditions, using coloring monodromy and connections to branched covers, Belyi surfaces, and cone-metrics.

Contribution

It introduces a coloring monodromy approach to analyze vertex degree conditions and extends the theory to branched covers and geometric structures.

Findings

Exceptional vertices are not adjacent when all but two have degrees divisible by k.

Coloring monodromy helps prove the case for k=2 and relates to branched covers.

Connections are established with Belyi surfaces and cone-metrics of constant curvature.

Abstract

If all but two vertices of a triangulated sphere have degrees divisible by $k$, then the exceptional vertices are not adjacent. This theorem is proved for $k=2$ with the help of the coloring monodromy. For $k = 3, 4, 5$ colorings by the vertices of platonic solids have to be used. With a coloring monodromy one can associate a branched cover. This generalizes to a space of germs between two triangulated surfaces. We also discuss relations with Belyi surfaces and with cone-metrics of constant curvature.