  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 $G$. If $G$ is not 3-edge-colorable, then $\mu_3(G) \geq 3$. In
[E. Steffen, 1-factor and cycle covers of cubic graphs, J. Graph Theory 78(3)
(2015) 195-206] it is shown that if $\mu_3(G) \not = 0$, then $2 \mu_3(G)$ is
an upper bound for the girth of $G$. We show that $\mu_3(G)$ bounds the oddness
$\omega(G)$ of $G$ as well. We prove that $\omega(G)\leq \frac{2}{3}\mu_3(G)$.
If $\mu_3(G) = \frac{2}{3} \mu_3(G)$, then every $\mu_3(G)$-core has a very
specific structure. We call these cores Petersen cores. We show that for any
given oddness there is a cyclically 4-edge-connected cubic graph $G$ with
$\omega(G) = \frac{2}{3}\mu_3(G)$. On the other hand, the difference between
$\omega(G)$ and $\frac{2}{3}\mu_3(G)$ can be arbitrarily big. This is true even
if we additionally fix the oddness. Furthermore, for every integer $k\geq 3$,
there exists a bridgeless cubic graph $G$ such that $\mu_3(G)=k$.
