Number of Edges in 3-Connected Graphs with Cyclic Neighborhoods

TL;DR

This paper provides a shorter proof for the minimum number of edges in 3-connected graphs where each vertex's neighborhood contains a cycle, advancing understanding of graph connectivity and cyclic neighborhoods.

Contribution

The paper offers a significantly shorter proof for the lower bound of edges in 3-connected graphs with cyclic neighborhoods, confirming the exact minimum of 15n/8 edges.

Findings

Every such graph has at least 15n/8 edges.

Examples exist with exactly 15n/8 edges.

Shorter proof of the existing bound.

Abstract

Chernyshev, Rauch and Rautenbach [Discrete Math., 2025] introduce forest cuts, i.e., vertex separators that induce a forest. They conjecture that, similar to a result by Chen and Yu [Discrete Math., 2002], every $n$-vertex graph with less than $3n-6$ edges has a forest cut. As an intermediate goal they ask how many edges an $n$-vertex $3$-connected graph must have such that the neighborhood of every vertex contains a cycle. Li, Tang and Zhan [arXiv, 2024] resolve this problem by showing that every such graph has at least $15n/8$ edges, while there are examples of such graphs with exactly $15n/8$ edges. We give a much shorter proof for this.