Abstract
Two 3-colorings of a cycle are complementary if whenever a vertex has its neighbors colored alike in one coloring, they are colored differently in the other coloring. Describing complementary colorings in terms of heawood colorings, we are able to count all such pairs. Complementary colorings can be defined for triangulations of manifolds. We construct complementary colorings for all oriented surfaces and for the 3-sphere. Finally, we apply these colorings to constructing triangulations whose odd part is a manifold.
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