## Illinois Journal of Mathematics

### A "nice" map colour theorem

K. S. Sarkaria

#### Abstract

A closed orientable triangulated surface is "nice" if its vertices can be assigned 4 colours in such a way that all 4 colours are used in the closed star of each edge. The 4-colouring can be interpreted as a simplicial map from the surface to the 4-vertex 2-sphere. If the surface has genus $(n-1)^2$, then the degree of this map is at least $n^2$. Conversely we show that, if $n$ is not divisible by 2 and 3, then there are "nice" surfaces of genus $(n-1)^2$ for which the degree of the above map is exactly $n^2$. Complex analytically "nice" surfaces can be viewed as minimally triangulated meromorphic functions of a Riemann surface.

#### Article information

Source
Illinois J. Math., Volume 46, Number 4 (2002), 1111-1123.

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258138469

Digital Object Identifier
doi:10.1215/ijm/1258138469

Mathematical Reviews number (MathSciNet)
MR1988253

Zentralblatt MATH identifier
1030.57037

#### Citation

Sarkaria, K. S. A "nice" map colour theorem. Illinois J. Math. 46 (2002), no. 4, 1111--1123. doi:10.1215/ijm/1258138469. https://projecteuclid.org/euclid.ijm/1258138469