Hamiltonian paths in Cayley digraphs of finitely-generated infinite abelian groups

2-generated Cayley digraphs on nilpotent groups have hamiltonian paths

Suppose G is a nilpotent, finite group. We show that if {a, b} is any 2-element generating set of G, then the corresponding Cayley digraph −−→ Cay(G; a, b) has a hamiltonian path. This implies that all of the connected Cayley graphs of valence ≤ 4 on G have hamiltonian paths.

Which Finitely Generated Abelian Groups Admit Isomorphic Cayley Graphs?

We show that Cayley graphs of finitely generated Abelian groups are rather rigid. As a consequence we obtain that two finitely generated Abelian groups admit isomorphic Cayley graphs if and only if they have the same rank and their torsion parts have the same cardinality. The proof uses only elementary arguments and is formulated in a geometric language.

Finitely Generated Abelian Groups

Cayley graphs of finitely generated groups

There does not exist a Borel choice of generators for each finitely generated group which has the property that isomorphic groups are assigned isomorphic Cayley graphs.

Automorphism groups and isomorphisms of Cayley digraphs of Abelian groups

Let S be a minimal generating subset of the finite abelian group G. We prove that if the Sylow 2-subgroup of G is cyclic, then Sand S U S-l are CI-subsets and the corresponding Cayley digraph and graph are normal. Let G be a finite group and let S be a subset of G not containing the identity element 1. The Cayley digraph X = Cay(G, S) of G with respect to S is defined by V(X) = G, E(X) = {(g,sg...