Equivelar Map

An equivelar map of type is a cellular embedding of a connected graph in a closed surface such that every face has edges and every vertex has vertex degree . If the map has vertices, edges, and faces, double counting incidences gives

Its Euler characteristic is therefore

For an orientable surface of genus , this equals .

Every regular map is an equivelar map, but an equivelar map need not be regular because flag transitivity is not required. The dual map of an equivelar map of type is equivelar of type . The five Platonic solids give the equivelar maps on the sphere that are also regular. The equivelar octahedron of genus 3 is a nonregular example of type .