Four colouring the vertices of the triangulation of a polygon containing a hole

Abstract

A simple linear-time algorithm is presented for four-colouring the vertices of a triangulation of a polygon containing a single hole. The algorithm consists of reducing a triangulation by the removal of both polygon and hole ear vertices, if any, followed by the removal of colour-isolated vertices until a 3-coloured tessellation remains. The triangulation is then re-built, using at most four colours. The paper concludes by recognising the similarity between the colouring of triangulations of polygons containing a hole and the colouring of bipartite and permutation graphs. © Springer-Verlag 2003.