A note on a conjecture concerning tree-partitioning 3-regular graphs

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