Maximal Embeddings of Directed Multi-Cycles

Abstract

We consider embeddings of Eulerian digraphs that have in-arcs alternating with out-arcs in the rotation schemes at each vertex. We define the multicycle Cnl,m to be the digraph on the vertex set v1,v2,ldots,vn, with arcs comprising l copies of the cycle (v1,v2,ldots,vn) and m copies of the cycle (vn,vn−1,ldots,v1). We consider maximal embeddings of multicycles and show that all except the bracelet digraphs Cn1,1 are upper-embeddable. We find that some multicycles have the maximum possible genus range, being both upper-embeddable and planar, and some multicycles have a genus range of zero.