Extremal Problems on Forest Cuts and Acyclic Neighborhoods in Sparse Graphs

TL;DR

Probes extremal edge-density conditions guaranteeing forest cuts in connected graphs, presenting a new $e$-density bound. The main result shows that if $G$ has fewer than $e(G) < \frac{9}{4}n$ edges, there exists a forest cut, tightening the previous bound of $e(G) < \frac{11}{5}n-\frac{18}{5}$. Additionally, the paper derives sharp lower bounds for 3-connected $1$-cyclic graphs ($m \ge \frac{15}{8}n$) and for 3-connected $2$-cyclic graphs ($m \ge 2n$), with matching constructions. It also frames an $\alpha$-FC Conjecture perspective, provides counterexamples to related conjectures, and discusses asymptotic tightness via blow-up schemes, offering a roadmap toward the conjectured $3n-6$ edge threshold.

Abstract

Chernyshev, Rauch, and Rautenbach proved that every connected graph on $n$ vertices with less than $\frac{11}{5}n-\frac{18}{5}$ edges has a vertex cut that induces a forest, and conjectured that the same remains true if the graph has less than $3n-6$ edges. We improve their result by proving that every connected graph on $n$ vertices with less than $\frac{9}{4}n$ edges has a vertex cut that induces a forest. We also study weaker versions of the problem that might lead to an improvement on the bound obtained.

Figures (2)

• Figure 1: A counterexample to Conjecture \ref{['conj:construction']} built from $K_4$.
• Figure 2: Left: Three 3-connected 2-cyclic graphs, two with 6 vertices and 12 edges and one with 9 vertices and 20 edges. Right: Three 4-connected 2-cyclic graphs, one with 7 vertices and 16 edges, and two with 8 vertices and 18 edges.

Theorems & Definitions (14)

• Conjecture 1: Chernyshev--Rauch--Rautenbach, 2024
• Conjecture 2: Chernyshev--Rauch--Rautenbach, 2024
• Theorem 3
• Remark 4
• Conjecture 5
• Conjecture 6: $\alpha$-FC Conjecture
• Lemma 7
• Corollary 8
• proof : Proof of Theorem \ref{['thm:teo']}\ref{['thm:9-4']}
• Lemma 9