## Illinois Journal of Mathematics

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

### A "nice" map colour theorem

#### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/ijm/1258138469

**Mathematical Reviews number (MathSciNet)**

MR1988253

**Zentralblatt MATH identifier**

1030.57037

**Subjects**

Primary: 57Q15: Triangulating manifolds

Secondary: 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25] 55M25: Degree, winding number

#### 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