Rotatieschemas

Rotatieschemas, also called rotation systems, are a formal way to describe how a finite graph can be drawn on a surface without edge crossings in a way that makes faces well-defined. A rotation system assigns to each vertex a cyclic order of the edges incident to that vertex. When combined with a pairing of edge-ends into whole edges, this local data determines a cellular embedding of the graph into an orientable surface (unless specified otherwise).

Formally, let G = (V,E) be a graph. For each v in V, choose a cyclic order of the half-edges incident to v. The collection of these orders is the rotation system. Together with the edge incidence, this data yields a surface embedding: traversing around a directed edge and using the cyclic order to turn at its head, then crossing to the opposite end, continues around a face. The faces obtained are the boundary cycles of the embedding.

From the embedding one computes the number of faces F; the Euler characteristic χ = V − E + F determines the genus: for orientable embeddings χ = 2 − 2g, so g = (2 − χ)/2. Non‑orientable cases are described by a related parameter. Planar graphs correspond to genus 0 embeddings, realized by rotation systems whose faces are topologically disks.

Rotatieschemas are central in topological graph theory and the theory of combinatorial maps. They provide a compact, purely combinatorial way to specify embeddings, to enumerate maps, and to study properties such as genus, orientability, and symmetry. Variants exist for non‑orientable embeddings, often modeled by extended permutation representations or by allowing flips in the local order.