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arXiv · 2605.24940

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

Béla Csaba
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Abstract

The Szemer\'edi 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.

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Béla Csaba. 2026-05-24. Decomposition of degree-regular graphs into quasi-random pairs without the Regularity lemma. https://arxiv.org/abs/2605.24940

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