Regular decomposition of the edge set of graphs with applications

TL;DR

This paper presents a novel edge decomposition method for graphs that can replace Szemerédi's Regularity lemma in certain embedding problems, along with an algorithmic implementation.

Contribution

It introduces a new graph edge decomposition technique and provides an algorithmic version, offering an alternative to the Regularity lemma for specific applications.

Findings

Effective in replacing the Regularity lemma in some embedding problems

Provides an algorithmic approach for the decomposition method

Enhances tools for graph embedding and analysis

Abstract

We introduce a new method for decomposing the edge set of a graph, and use it to replace the Regularity lemma of Szemer\'edi in some graph embedding problems. An algorithmic version is also given.