[Paper Review] A simple regularization of graphs

Yoshiyasu Ishigami|ArXiv.org|Apr 30, 2009

Limits and Structures in Graph Theory11 references3 citations

TL;DR

This paper presents a novel, non-iterative construction of vertex partitions in graphs that satisfy quasi-randomness properties via a single random sampling of vertices, using a randomly chosen sample size from a pre-defined sequence. The method ensures high-probability regularization with constant-sized partitions, offering a simpler, iteration-free alternative to Szemerédi’s original energy-increment approach, and generalizes naturally to hypergraphs.

ABSTRACT

The well-known regularity lemma of E. Szemerédi for graphs (i.e. 2-uniform hypergraphs) claims that for any graph there exists a vertex partition with the property of quasi-randomness. We give a simple construction of such a partition. It is done just by taking a constant-bounded number of random vertex samplings only one time (thus, iteration-free). Since it is independent from the definition of quasi-randomness, it can be generalized very naturally to hypergraph regularization. In this expository note, we show only a graph case of the paper [I] on hypergraphs, but may help the reader to access [I].

Motivation & Objective

• To provide a simple, non-iterative construction of vertex partitions that satisfy quasi-randomness in graphs.
• To overcome the limitations of iterative energy-increment methods in Szemerédi’s regularity lemma by using a single random sampling procedure.
• To present a framework that is independent of the specific definition of quasi-randomness, enabling natural generalization to hypergraphs.
• To offer a more accessible entry point to the broader hypergraph regularity theory developed in a companion paper [5].
• To demonstrate that a single random sampling with a randomly selected sample size yields a partition with high probability satisfying the counting condition for quasi-randomness.

Proposed method

• A large constant $\tilde{n}$ is chosen based on the desired error tolerance $\epsilon$, independent of the input graph.
• A sequence of increasing sample sizes $m_0 \ll m_1 \ll \cdots \ll m_{\tilde{n}-1}$ is defined, also independent of the graph.
• An integer $n \in \{0, \dots, \tilde{n}-1\}$ is selected uniformly at random, and $m_n$ vertices are sampled uniformly at random from the graph.
• Each vertex is labeled by its adjacency pattern to the $m_n$ sampled vertices, forming a partition into at most $2^{m_n} \leq 2^{m_{\tilde{n}-1}}$ parts.
• The resulting partition is shown to satisfy the quasi-randomness condition based on induced subgraph counts with high probability.
• The proof relies on telescoping expectations and concentration bounds, avoiding iterative refinement or energy increment arguments.

Experimental results

Research questions

• RQ1 Can a vertex partition satisfying quasi-randomness be constructed without iterative refinement, using only a single random sampling step?
• RQ2 Is it possible to achieve high-probability regularization in graphs using a method independent of the specific definition of quasi-randomness?
• RQ3 How can the construction be generalized to hypergraphs while preserving simplicity and correctness?
• RQ4 What is the minimal number of random samples needed to ensure quasi-randomness in the partition?
• RQ5 Does a single random sampling with a random sample size yield a partition that satisfies the counting lemma for induced subgraphs?

Key findings

• The proposed method constructs a vertex partition with constant-sized parts that satisfies quasi-randomness with high probability using only one random sampling step.
• The construction avoids iterative refinement, unlike Szemerédi’s original energy-increment method, and is independent of the definition of quasi-randomness.
• The method generalizes naturally to hypergraphs, as shown in the companion paper [5], and provides a unified framework for hypergraph regularity.
• The expected regularity of the partition is bounded by $\epsilon$, ensuring that the partition is $\epsilon$-regular with high probability.
• The proof relies on telescoping expectations and concentration inequalities, showing that the error in induced subgraph counts is controlled by $\epsilon$.
• The approach achieves the same goal as the regularity lemma with significantly simpler algorithmic structure and lower computational overhead.