On normality of n-Cayley graphs

摘要

Let G be a finite group and X a (di)graph. If there exists a semiregular subgroup G¯ of the automorphism group  Aut(X) isomorphic to G with n orbits on V(X) then the (di)graph X is called an n-Cayley graph on G. If, in addition, this subgroup G¯ is normal in  Aut(X) then X is called a normal n-Cayley graph on G.In this paper the normalizers of semiregular subgroups of the automorphism group of a digraph are characterized. It is proved that every finite group admits a vertex-transitive normal n-Cayley graph for every n ≥ 2. For the most part the graphs are constructed as Cartesian product of graphs. It is proved that a Cartesian product of two relatively prime graphs is Cayley (resp. normal Cayley) if and only if the factor graphs are Cayley (resp. normal Cayley). In addition, the concept of graphical regular representations (GRRs) is generalized to n-GRR in a natural way, and it is proved that any group admitting a GRR also admits an n-GRR for any n ≥ 1.