Given a graph $G$, let $H$ be a $k$-degenerate (not necessarily induced) subgraph of maximal size. Are there any known lower bounds on $|E(H)|$ for particular classes of $G$ and values of $k$?

I've seen several papers on the subject, but only for *induced* subgraphs $H$. I'm particularly interested in the case where $k=2$ and $G$ is both regular and bipartite, but any additional information would be helpful.

(I'm aware of the naive lower bound found by arbitrarily removing edges until no vertex has degree $(k+1)$ or more...I'm looking for something better).