  We show that a cubic graph $G$ of order $n$ has an induced $2$-regular
subgraph of order at least a) $\frac{n-2}{4-\frac{4}{k}}$, if $G$ has no
induced cycle of length more than $k$, b) $\frac{5n+6}{8}$, if $G$ has no
induced cycle of length more than $4$, and $n>6$, and c)
$\left(\frac{1}{4}+\epsilon\right)n$, if the independence number of $G$ is at
most $\left(\frac{3}{8}-\epsilon\right)n$. To show the second result we give a
precise structural description of cubic $4$-chordal graphs.
