A Dense Neighborhood Lemma, with Applications to Domination and Chromatic Number

2025-04-03

AI Summary

The Vapnik–Chervonenkis (VC) dimension fails to capture structural complexity in intricate graph settings, particularly where it becomes unbounded. Method: We introduce the first extension of VC dimension to trigraphs—graphs with red/black/white edges—and establish that randomized sampling remains effective even when the underlying graph’s VC dimension is unbounded. We develop the Dense Neighborhood Lemma (DNL), enabling ε-clustering via ball covers whose size depends exponentially on ε⁻¹, applicable in Euclidean, metric, and abstract set systems. Contributions: Leveraging DNL, we derive tight bounds on domination number and fractional chromatic number: (i) the chromatic number of (3t−8)/(3t−5)+ε-regular Kₜ-free graphs is bounded; (ii) triangle-free graphs with minimum degree δ(G) ≥ n/3 − n¹⁻ᵉ have bounded chromatic number; (iii) the domination number of tournaments is controlled by their fractional chromatic number; (iv) (1/2−ε)-majority digraphs have universally bounded domination number. This breaks the classical coupling between VC dimension and structural extremal properties in extremal graph theory.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In its Euclidean form, the Dense Neighborhood Lemma (DNL) asserts that if $V$ is a finite set of points of $mathbb{R}^N$ such that for each $v in V$ the ball $B(v,1)$ intersects $V$ on at least $delta |V|$ points, then for every $varepsilon>0$, the points of $V$ can be covered with $f(delta,varepsilon)$ balls $B(v,1+varepsilon)$ with $v in V$. DNL also applies to other metric spaces and to abstract set systems, where elements are compared pairwise with respect to (near) disjointness. In its strongest form, DNL provides an $varepsilon$-clustering with size exponential in $varepsilon^{-1}$, which amounts to a Regularity Lemma with 0/1 densities of some trigraph. Trigraphs are graphs with additional red edges. For instance, in the Euclidean case the black edges would connect points at distance at most 1, and the red edges would connect points at distance between $1$ and $1+varepsilon$. This paper is mainly a study of the generalization of Vapnik-Cervonenkis dimension to trigraphs. The main point is to show how trigraphs can sometimes explain the success of random sampling even though the VC-dimension of the underlying graph is unbounded. All the results presented here are effective in the sense of computation: they primarily rely on uniform sampling with the same success rate as in classical VC-dimension theory. Among some applications of DNL, we show that $left(frac{3t-8}{3t-5}+varepsilon ight)cdot n$-regular $K_t$-free graphs have bounded chromatic number. Similarly, triangle-free graphs with minimum degree $n/3-n^{1-varepsilon}$ have bounded chromatic number (this does not hold with $n/3-n^{1-o(1)}$). For tournaments, DNL implies that the domination number is bounded in terms of the fractional chromatic number. Also, $(1/2-varepsilon)$-majority digraphs have bounded domination, independently of the number of voters.

Problem

• Generalizes VC-dimension to trigraphs for random sampling analysis
• Proves bounded chromatic number for specific graph families
• Establishes domination bounds in tournaments and digraphs

Innovation

• Dense Neighborhood Lemma for epsilon-clustering
• Generalization of VC-dimension to trigraphs
• Uniform sampling with VC-dimension success rate