Color or cover

2015/03/02 by Ivan Izmestiev, Izmestiev, Ivan
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #math.CO #math.GT

paper · pdf · doi:10.48550/arxiv.1503.00605

12 pages, 8 figures; references to Steve Fisk added

arxiv created 2015/11/20 · arxiv updated 2015/11/23

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.

Related

• Continued Fractions and the 4-Color Theorem
• Non-compact Riemann surfaces are equilaterally triangulable
• Extending triangulations of the 2-sphere to the 3-disk preserving a 4-coloring
• Triangulated spheres with holes in triangulated surfaces
• The surface of a sufficiently large sphere has chromatic number at most\n 7