Illinois Journal of Mathematics
- Illinois J. Math.
- Volume 46, Number 4 (2002), 1111-1123.
A "nice" map colour theorem
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.
Illinois J. Math., Volume 46, Number 4 (2002), 1111-1123.
First available in Project Euclid: 13 November 2009
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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