  An arc of a graph is an oriented edge and a 3-arc is a 4-tuple $(v,u,x,y)$ of
vertices such that both $(v,u,x)$ and $(u,x,y)$ are paths of length two. The
3-arc graph of a graph $G$ is defined to have vertices the arcs of $G$ such
that two arcs $uv, xy$ are adjacent if and only if $(v,u,x,y)$ is a 3-arc of
$G$. In this paper we prove that any connected 3-arc graph is Hamiltonian, and
all iterative 3-arc graphs of any connected graph of minimum degree at least
three are Hamiltonian. As a consequence we obtain that if a vertex-transitive
graph is isomorphic to the 3-arc graph of a connected arc-transitive graph of
degree at least three, then it is Hamiltonian. This confirms the well known
conjecture, that all vertex-transitive graphs with finitely many exceptions are
Hamiltonian, for a large family of vertex-transitive graphs. We also prove that
if a graph with at least four vertices is Hamilton-connected, then so are its
iterative 3-arc graphs.
