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A Dense Neighborhood Lemma: Applications of Partial Concept Classes to Domination and Chromatic Number

Romain Bourneuf, Pierre Charbit, Stéphan Thomassé

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 $δ|V|$ points, then for every $\varepsilon >0$, the points of $V$ can be covered with $f(δ,\varepsilon)$ 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. They are natural instances of partial concept classes, introduced by Alon, Hanneke, Holzman and Moran [FOCS 2021]. This paper is mainly a combinatorial study of the generalization of Vapnik-Cervonenkis dimension to partial concept classes. 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 $\left(\frac{3t-8}{3t-5}+\varepsilon\right)\cdot n$-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.

A Dense Neighborhood Lemma: Applications of Partial Concept Classes to Domination and Chromatic Number

Abstract

In its Euclidean form, the Dense Neighborhood Lemma (DNL) asserts that if is a finite set of points of such that for each the ball intersects on at least points, then for every , the points of can be covered with balls with . 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 -clustering with size exponential in , which amounts to a Regularity Lemma with 0/1 densities of some trigraph. Trigraphs are graphs with additional red edges. They are natural instances of partial concept classes, introduced by Alon, Hanneke, Holzman and Moran [FOCS 2021]. This paper is mainly a combinatorial study of the generalization of Vapnik-Cervonenkis dimension to partial concept classes. 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 -regular -free graphs have bounded chromatic number. Similarly, triangle-free graphs with minimum degree have bounded chromatic number (this does not hold with ). For tournaments, DNL implies that the domination number is bounded in terms of the fractional chromatic number. Also, -majority digraphs have bounded domination, independently of the number of voters.

Paper Structure

This paper contains 43 sections, 72 theorems, 58 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Examples of Dense Neighborhood Lemma
  3. VC-dimension of Trigraphs
  4. Tri-hypergraphs
  5. Domination in Trigraphs
  6. Clustering
  7. Chromatic thresholds of Kt-free graphs
  8. Domination versus fractional chromatic number in tournaments
  9. Domination in majority directed graphs
  10. Preliminaries
  11. Set systems.
  12. Graphs.
  13. Trigraphs.
  14. Trigraph parameters.
  15. Classes of trigraphs.
  16. ...and 28 more sections

Key Result

Lemma 0

Let $V$ be a finite subset of $\mathbb{R}^N$. If every $V$-ball of radius 1 intersects $V$ on at least $\delta |V|$ points, then for all $\varepsilon >0$, the set $V$ can be covered by $\textup{poly}(\delta^{-1},\varepsilon^{-1})$$V$-balls with radius $1+\varepsilon$.

Figures (5)

  • Figure 1: Edges of $\mathcal{I}$
  • Figure 2: Edge of $\mathcal{D}$ which is the difference between the tri-edge corresponding to the the neighborhood of $v$ in $T_1$ and the tri-edge corresponding to the neighborhood of $w$ in $T_2$. Observe that every vertex in the black zone is at distance at most $\tau N$ of $v$ and at least $(\tau + \varepsilon)N$ of $w$, and every vertex in the white zone is either at distance at least $(\tau + \varepsilon/2) N$ of $v$ or at distance at most $(\tau + \varepsilon/2) N$ of $w$.
  • Figure 3: Brandt's graph.
  • Figure 4: O'Rourke's construction of a $(7/10 - \varepsilon)N$-regular $K_5$-free $N$-vertex graph with large chromatic number. The graph $B$ is a Borsuk graph, triangle-free with large chromatic number. The neighborhoods of the vertices of $I$ in $B$ give a $2 + \varepsilon'$-fractional coloring of $B$.
  • Figure 5: Setup for the end of the proof of \ref{['thm:regularkt']}.

Theorems & Definitions (78)

  • Lemma 0
  • Lemma 0
  • Lemma 0
  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 4
  • Lemma 5
  • ...and 68 more