If *A* is a square-free subset of an abelian group *G*, then the
addition graph of *A*
on *G* is the graph with vertex set *G* and distinct vertices *x* and
*y* forming an edge if and only if *x+y* is in *A*. We prove that every
connected cubic addition graph on an abelian group *G* whose
order is divisible by 8 is Hamiltonian as well as every
connected bipartite cubic addition graph on an abelian group *G* whose
order is divisible by 4. We show that connected bipartite addition
graphs are Cayley graphs and prove that every connected cubic Cayley
graph on a group of dihedral type whose
order is divisible by 4 is Hamiltonian.