# Regular Maps of High Density

Discrete & Computational Geometry, Mar 2017

A regular map is a surface together with an embedded graph, having properties similar to those of the surface and graph of a platonic solid. We analyze regular maps with reflection symmetry and a simple graph with ratio between vertex-degree and number of vertices strictly exceeding $\frac{1}{2}$. We conclude that all regular maps of this type belong to a family of maps naturally defined on the Fermat curves $x^n+y^n+z^n=0$, excepting the one corresponding to the tetrahedron.

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs00454-017-9879-6.pdf

Rob H. Eggermont, Maximiliaan Hendriks. Regular Maps of High Density, Discrete & Computational Geometry, 2017, 881-895, DOI: 10.1007/s00454-017-9879-6