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.