Decomposition of degree-regular graphs into quasi-random pairs without the Regularity lemma

Abstract

The Szemerédi Regularity Lemma, in combination with the Blow-up Lemma, form the Regularity Method, a fundamental tool in graph embeddings, albeit restricted to very large and dense graphs. We propose an alternative vertex-partitioning framework that remains effective even as the density tends to zero and without requiring astronomically large vertex sets. This approach, while narrower in scope, extends regularity-type techniques to relatively small graphs previously inaccessible to the Regularity Method. As an application, we use this novel vertex-partitioning method for bipartite packing problems.