We say that a $2$-coloring of a graph $G$ is unfriendly if every vertex has at least as many neighbors of the color opposite to his own. Such a coloring always exist and can be obtained by partitioning the graph into two sets that maximize the number of edges between them.
Specifically for a cubic graph $G$ a unfriendly coloring is a partition $(A,B)$ of $V(G)$ so that every vertex in $A$ has at least two neighbors in $B$ and vice versa.
We say that a cubic graph is imbalanced if it has a unfriendly partition where the two parts have different sizes.
For example, $K_4$ is not imbalanced and so isn't the cube graph $Q_3$.
Out of $1413230$ cubic planar graphs on at most $24$ vertices, there are precisely $2160$ graphs that are not imbalanced and they are all Hamiltonian. Similarly all snarks on up to $32$ vertices are imbalanced.
Hence I wonder
Is every cubic non-Hamiltonian graph imbalanced?
I would like to understand the notion of imbalanced a little bit better, so any other structural remarks about such graphs are also welcome.