Nonoverlapping partitions of a surface ^*

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 ≥ 3 and nonorientable surfaces of genus g ≥ 4.