A Note on Large Degenerate Induced Subgraphs in Sparse Graphs

TL;DR

This paper establishes improved lower bounds on the size of large induced d-degenerate subgraphs in sparse graphs, extending previous results and applying to graphs of bounded genus.

Contribution

It provides new lower bounds for induced d-degenerate subgraphs in k-degenerate and bounded genus graphs, improving upon prior bounds.

Findings

Improved lower bounds for induced d-degenerate subgraphs in k-degenerate graphs.

Extended bounds to graphs of bounded genus.

Generalizes earlier results on planar graphs.

Abstract

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 argument we obtain improved lower bound on $\alpha_d(G)$ for graphs of bounded genus. This extends earlier work on degenerate subgraphs of planar graphs.