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.
Citation
K. S. Sarkaria. "A "nice" map colour theorem." Illinois J. Math. 46 (4) 1111 - 1123, Winter 2002. https://doi.org/10.1215/ijm/1258138469
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