Two families of graphs that are Cayley on nonisomorphic groups

A number of authors have studied the question of when a graph can be represented as a Cayley graph on more than one nonisomorphic group. The work to date has focussed on a few special situations: when the groups are $p$-groups; when the groups have order $pq$; when the Cayley graphs are normal; or when the groups are both abelian. In this paper, we construct two infinite families of graphs, each of which is Cayley on an abelian group and a nonabelian group. These families include the smallest examples of such graphs that had not appeared in other results.Comment: 6 page

On isomorphisms of abelian Cayley objects of certain orders

AbstractLet m be a positive integer such that gcd(m,ϕ(m))=1 (ϕ is Euler's phi function) with m=p1⋯pr the prime power decomposition of m. Let n=p1a1⋯prar. We provide a sufficient condition to reduce the Cayley isomorphism problem for Cayley objects of an abelian group of order n to the prime power case. In the case of Cayley k-ary relational structures (which include digraphs) of abelian groups, this sufficient condition reduces the Cayley isomorphism problem of k-ary relational structures of abelian groups to the prime power case for Cayley k-ary relational structures of abelian groups. As corollaries, we solve the Cayley isomorphism problem for Cayley graphs of Zn (for the specific values of n as above) and prove several abelian groups (for specific choices of the ai) of order n are CI-groups with respect to digraphs