  Let $\mathcal{F}$ be a family of fixed graphs and let $d$ be large enough.
For every $d$-regular graph $G$, we study the existence of a spanning
$\mathcal{F}$-free subgraph of $G$ with large minimum degree. This problem is
well-understood if $\mathcal{F}$ does not contain bipartite graphs. Here we
provide asymptotically tight results for many families of bipartite graphs such
as cycles or complete bipartite graphs.
