  A $k$-weak bisection of a cubic graph $G$ is a partition of the vertex-set of
$G$ into two parts $V_1$ and $V_2$ of equal size, such that each connected
component of the subgraph of $G$ induced by $V_i$ ($i=1,2$) is a tree of at
most $k-2$ vertices. This notion can be viewed as a relaxed version of
nowhere-zero flows, as it directly follows from old results of Jaeger that
every cubic graph $G$ with a circular nowhere-zero $r$-flow has a $\lfloor r
\rfloor$-weak bisection. In this paper we study problems related to the
existence of $k$-weak bisections. We believe that every cubic graph which has a
perfect matching, other than the Petersen graph, admits a 4-weak bisection and
we present a family of cubic graphs with no perfect matching which do not admit
such a bisection. The main result of this article is that every cubic graph
admits a 5-weak bisection. When restricted to bridgeless graphs, that result
would be a consequence of the assertion of the 5-flow Conjecture and as such it
can be considered a (very small) step toward proving that assertion. However,
the harder part of our proof focuses on graphs which do contain bridges.
