A Note on Large Degenerate Induced Subgraphs in Sparse Graphs

TL;DR

This work studies the size of induced $d$-degenerate subgraphs in sparse graphs through the invariant $\alpha_d(G)$. It proves a central bound for $k$-degenerate graphs: $\alpha_d(G) \ge \max\{ \frac{(d+1)n}{k+d+1}, n-\alpha_{k-d-1}(G) \}$, and shows a vertex partition into a $d$-degenerate part and a $(k-d-1)$-degenerate part. The authors extend these ideas to graphs on surfaces of bounded genus, obtaining explicit lower bounds for $\alpha_i(G)$ across several values of $i$, and they discuss limitations of acyclic-coloring-based approaches in this setting. Together these results advance lower bounds beyond the classic Alon–Kahn–Seymour bound for $k$-degenerate graphs and broaden the understanding of degenerate induced subgraphs beyond planar graphs, while outlining computational and conjectural directions for future work.

Abstract

Given a graph $G$ and a non-negative integer $d$ let $α_d(G)$ 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 $α_d(G) \geq \max\{ \frac{(d+1)n}{k+d+1}, n - α_{k-d-1}(G)\}$. 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 $α_d(G)$ for graphs of bounded genus. This extends earlier work on degenerate subgraphs of planar graphs.

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 1.2
• proof : Proof of Theorem \ref{['Thm: Main Degen']}
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• proof : Proof of Theorem \ref{['Thm: Surfaces']}
• Conjecture 4.1
• Conjecture 4.2