If G is a 4-connected maximal planar graph, then G is hamiltonian (by a theorem\ud
of Whitney), implying that its dual graph G� is a cyclically 4-edge connected 3-\ud
regular planar graph admitting a partition of the vertex set into two parts, each\ud
inducing a tree in G�, a so-called tree-partition. It is a natural question whether\ud
each cyclically 4-edge connected 3-regular graph admits such a tree-partition.\ud
This was conjectured by Jaeger, and recently independently by the �rst author.\ud
The main result of this note shows that each connected 3-regular graph on n\ud
vertices admits a partition of the vertex set into two sets such that precisely\ud
12\ud
n+2 edges have end vertices in each set. This is a necessary condition for having\ud
a tree-partition. We also show that not all cyclically 3-edge connected 3-regular\ud
(planar) graphs admit a tree-partition, and present the smallest counterexample
