Number of Edges in 3-Connected Graphs with Cyclic Neighborhoods

2025-11-13

AI Summary

This paper determines the minimum number of edges in a 3-connected $n$-vertex graph where every vertex neighborhood contains a cycle (“locally cyclic”). Leveraging combinatorial analysis and classical graph-theoretic techniques—particularly those exploiting 3-connectivity and structural constraints on neighborhoods—we establish a tight lower bound of $frac{15}{8}n$. Our proof is self-contained and significantly simpler than the prior argument by Li–Tang–Zhan, while also improving the constant. This result provides crucial support for the Forest Decomposition Conjecture and advances the understanding of local structure in sparse 3-connected graphs. Moreover, we construct an infinite family of extremal graphs achieving this bound, thereby confirming its tightness.

Technology Category

• Search and Optimization: Local Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Bridging structured and unstructured data

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.

Problem

• Determining minimum edges in 3-connected graphs with cyclic neighborhoods
• Proving graphs need at least 15n/8 edges when neighborhoods contain cycles
• Providing shorter proof for edge threshold in cyclic neighborhood graphs

Innovation

• Introduces forest cuts as vertex separators
• Proves graphs need at least 15n/8 edges
• Provides shorter proof for 3-connected graphs