  Let $G$ be a bridgeless cubic graph. Consider a list of $k$ 1-factors of $G$.
Let $E_i$ be the set of edges contained in precisely $i$ members of the $k$
1-factors. Let $\mu_k(G)$ be the smallest $|E_0|$ over all lists of $k$
1-factors of $G$.
  Any list of three 1-factors induces a core of a cubic graph. We use results
on the structure of cores to prove sufficient conditions for Berge-covers and
for the existence of three 1-factors with empty intersection. Furthermore, if
$\mu_3(G) \not = 0$, then $2 \mu_3(G)$ is an upper bound for the girth of $G$.
We also prove some new upper bounds for the length of shortest cycle covers of
bridgeless cubic graphs.
  Cubic graphs with $\mu_4(G) = 0$ have a 4-cycle cover of length $\frac{4}{3}
|E(G)|$ and a 5-cycle double cover. These graphs also satisfy two conjectures
of Zhang. We also give a negative answer to a problem of Zhang.
