Complementary colorings (Q1339861)

Two 3-colourings of a cycle are complementary if, whenever a vertex has its neighbours coloured alike in one colouring, they are coloured differently in the other colouring. A Heawood colouring of a cycle is an assignment of \(\pm 1\) to each of the edges of the cycle. The author proves that every pair of complementary 3-colourings of an even cycle is induced by a pair of Heawood colourings. This enables him to count all pairs of complementary 3-colourings of a cycle. Complementary 4- colourings of surfaces are defined in an analogous manner, and complementary colourings are constructed for all oriented surfaces and for the 3-sphere. These colourings are then applied to constructing triangulations whose odd part is a manifold.