Abstract: In this paper the authors investigate the embeddings of the circular graphs. The authors determine the minimum orientable genus and the minimum nonorientable genus and show that all the circular graphs are up-embeddable. The authors show that for a fixed integer $l (\geq 3)$ and large enough $n$, there is only one way to embed a 4-regular circular graph $C(n,l)$ into the torus such that each face is a quadrilateral. In particular, the authors find that both the torus and the Klein bottle may be quadrangulated by the circular graph $C(2l+2,l)$ which, by introducing some new edges, may also triangulate both of the two surfaces.

Key words: Circular graph, Embedding, Minimum (non-orientable) genus