A new graph decomposition method for bipartite graphs

TL;DR

This paper introduces a new graph decomposition technique for bipartite graphs that partitions most edges into quasirandom subgraphs, applicable to sparse graphs and useful for graph embedding tasks.

Contribution

It presents a novel bipartite graph decomposition method that replaces the Regularity lemma in certain embedding problems, effective for sparse and dense graphs.

Findings

Decomposes most edges into quasirandom subgraphs

Applicable to small and sparse bipartite graphs

Can substitute the Regularity lemma in some applications

Abstract

Given a sufficiently large and sufficiently dense bipartite graph $G=(A, B; E),$ we present a novel method for decomposing the majority of the edges of $G$ into quasirandom graphs so that the vertex sets of these quasirandom graphs partition the majority of $A.$ The method works for relatively small or sparse graphs, and can be used to substitute the Regularity lemma of Szemer\'edi in some graph embedding problems.