Degenerate Vertex Cuts in Sparse Graphs

TL;DR

This work studies the existence and size of $k$-degenerate vertex cuts in sparse graphs, where a cut induces a $k$-degenerate subgraph. Using $k$-core structure and discharging techniques, the authors prove a general lower bound $|E(G)| \ge \frac{1}{2}\left(k+\frac{13\sqrt{k}}{190}-\frac{1}{38}\right)n$ for graphs of order $n\ge 2k+2$ with no $k$-degenerate cut, and a sharper bound $|E(G)| \ge \frac{27n-35}{10}$ when $k=2$ and $n\ge 5$, along with a result guaranteeing a minimum $k$-degenerate cut in connected graphs with $n\ge k+6$ and $m\le \frac{k+3}{2}n+\frac{k-1}{2}$. The proofs combine local-structure arguments around low-degree vertices, $k$-core arguments, and discharging, plus a minimal-counterexample approach for the minimum $k$-degenerate cut result. The findings extend the landscape beyond independent and forest cuts, and the constructed extremal examples show the bounds are tight up to polylogarithmic factors in $k$.

Abstract

For a non-negative integer $k$, a vertex cut in a graph is $k$-degenerate if it induces a $k$-degenerate subgraph. We show that a graph of order $n$ at least $2k+2$ without a $k$-degenerate cut has the size at least $\frac{1}{2}\left(k+Ω\left(\sqrt{k}\right)\right)n$ and that a graph of order $n$ at least $5$ without a $2$-degenerate cut has the size at least $\frac{27n-35}{10}$. For $k\geq 2$, we show that a connected graph $G$ of order $n$ at least $k+6$ and size $m$ at most $\frac{k+3}{2}n+\frac{k-1}{2}$ has a minimum $k$-degenerate cut.

Figures (1)

• Figure 1: The constructed graph $G$ for $k=2$, $s=12$, and a specific choice of the matchings $M_i$.

Theorems & Definitions (12)

• Theorem 1
• Theorem 2
• Theorem 3
• proof : Proof of Theorem \ref{['thm:size']}
• Claim 1
• proof : Proof of Claim \ref{['claim1']}.
• proof : Proof of Theorem \ref{['thm2']}
• Claim 2
• proof : Proof of Claim \ref{['claim2']}.
• Lemma 4