Regular Triangulations of Non-Compact Surfaces

Abstract

A triangulation of a surface is $\delta$-regular if each vertex is contained in exactly $\delta$ edges. For each $\delta \ge 7$, $\delta$-regular triangulations of arbitrary non-compact surfaces of finite genus are constructed. It is also shown that for $\delta \le 6$ there is a $\delta$-regular triangulation of a non-compact surface $\sum$ if and only if $\delta =6$ and $\sum$ is homeomorphic to one of the following surfaces: the Euclidean plane, the two-way-infinite cylinder, or the open Möbius band.