Nonoverlapping partitions of a surface

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 neighbourhood 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 additional 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 surface.  The chromatic number is derived for both orientable surfaces of genus $g\ge 3$ and nonorientable surfaces of genus $g\ge 4$.