Nonoverlapping partitions of a surface

Simon Davis

doi:10.48165/

Authors

Simon Davis Research Foundation of Southern California, 8861 Villa La Jolla Drive #13595, La Jolla, CA 92039, U.S.A.

DOI:

https://doi.org/10.48165/

Keywords:

planar domains, ribboned regions, isometric circles, minimal number, Schottky problem, Riemann surfaces

Abstract

The colouring of planar domains is considered through the tight packing of rectangular regions. It is demonstrated that a maximal number of colours in a neigh bourhood is achieved through the introduction of ribboned regions. This number can be reduced to four in the brick model with a special choice of colours in the surrounding region. An exceptional planar domain found by interweaving a ribboned region with a compact hexagonal configuration of isometric circles of a Schottky group requires an ad ditional colour. The equivalent tight packing of isometric circles of the Schottky group provides a method for deriving the number of colours required to cover a Riemann sur face. The chromatic number is derived for both orientable surfaces of genus g ≥ 3 and nonorientable surfaces of genus g ≥ 4.

References

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