Imbeddings of metacyclic cayley graphs

Abstract

Any group whose commutator subgroup and commutator quotient group are both cyclic is called metacyclic. The Cayley graph corresponding to any generating set for a metacyclic group is called a metacyclic Cayley graph. Whereas the earliest work on Cayley graph imbeddings concentrates mainly on planarity, recent work of A.T. White and others shows that higher genus imbeddings are accessible, especially for abelian groups. Since metacyclic groups are an especially tractable kind of nonabelian groups, this paper beings a development of an imbedding theory for nonabelian Cayley graphs by considering the metacyclic Cayley graphs. The author has previously used voltage graphs (the duals of the current graphs of Gustin, Ringel, and Youngs) to show that a special subclass of metacyclic groups has toroidal Cayley graphs. In this paper, toroidal Cayley graphs are constructed for two additional subclasses of metacyclic groups.