A Note on Large Degenerate Induced Subgraphs in Sparse Graphs

Given a graph $G$ and a non-negative integer $d$ let $\alpha_d(G)$ be the order of a largest induced $d$-degenerate subgraph of $G$. We prove that for any pair of non-negative integers $k>d$, if $G$ is a $k$-degenerate graph, then $\alpha_d(G) \geq \max\{ \frac{(d+1)n}{k+d+1}, n - \alpha_{k-d-1}(G)\}$. For $k$-degenerate graphs this improves a more general lower bound of Alon, Kahn, and Seymour. By modifying our arguments we also obtain an improved lower bounds on $\alpha_d(G)$ for graphs of bounded genus. This extends earlier work on degenerate subgraphs of planar graphs.