A more formal definition of a rotation system involves pairs of permutations; such a pair is sufficient to determine a multigraph, a surface, and a 2-cell embedding of the multigraph onto the surface.
Every rotation scheme defines a unique 2-cell embedding of a connected multigraph on a closed oriented surface (up to orientation-preserving topological equivalence).
This fundamental equivalence between rotation systems and 2-cell-embeddings was first settled in a dual form by Lothar Heffter in the 1890s[1] and extensively used by Ringel during the 1950s.
[2] Independently, Edmonds gave the primal form of the theorem[3] and the details of his study have been popularized by Youngs.
A rotation system specifies a circular ordering of the edges around each vertex, while a rotation map specifies a (non-circular) permutation of the edges at each vertex.
To derive a rotation system from a 2-cell embedding of a connected multigraph G on an oriented surface, let B consist of the darts (or flags, or half-edges) of G; that is, for each edge of G we form two elements of B, one for each endpoint of the edge.
We let θ(b) be the other dart formed from the same edge as b; this is clearly an involution with no fixed points.
If a rotation system is derived from a 2-cell embedding of a connected multigraph G, the graph derived from the rotation system is isomorphic to G. To embed the graph derived from a rotation system onto a surface, form a disk for each orbit of σθ, and glue two disks together along an edge e whenever the two darts corresponding to e belong to the two orbits corresponding to these disks.
The result is a 2-cell embedding of the derived multigraph, the two-cells of which are the disks corresponding to the orbits of σθ.
According to the Euler formula we can deduce the genus g of the closed orientable surface defined by the rotation system