On the enumeration of circular maps with given number of edges

A map is a closed Riemann surface $S$ with an embedded graph $G$ such that $S\setminus G$ is homeomorphic to a disjoint union of open disks. Tutte began a systematic study of maps in the 1960s, and contemporary authors are actively developing it. We introduce the concept of circular map and establish its equivalence to the concept of map admitting a coloring of the faces in two colors. The main result is a formula for the number of circular maps with given number of edges.