Regular bipartite decompositions of pseudorandom graphs

TL;DR

This paper extends Kotzig's decomposition result from complete graphs to a broader class of pseudorandom graphs, showing they can be decomposed into a small number of regular bipartite subgraphs, with implications for graph factorizations.

Contribution

It introduces a randomized algorithm for decomposing $(n,d,)$-graphs into near-optimal regular bipartite subgraphs, generalizing Kotzig's classical result.

Findings

Decomposition into at most ()  d + 36 regular bipartite subgraphs.

Improved bounds on  for -graphs to admit 1-factorizations.

Decomposition results are tight up to an additive constant.

Abstract

In 1972, Kotzig proved that for every even $n$, the complete graph $K_n$ can be decomposed into $\lceil\log_2n\rceil$ edge-disjoint regular bipartite spanning subgraphs, which is best possible. In this paper, we study regular bipartite decompositions of $(n,d,\lambda)$-graphs, where $n$ is an even integer and $d_0\leq d\leq n-1$ for some absolute constant $d_0$. With a randomized algorithm, we prove that such an $(n,d,\lambda)$-graph with $\lambda\leq d/12$ can be decomposed into at most $\log_2 d + 36$ regular bipartite spanning subgraphs. This is best possible up to the additive constant term. As a consequence, we also improve the best known bounds on $\lambda = \lambda(d)$ by Ferber and Jain (2020) to guarantee that an $(n,d,\lambda)$-graph on an even number of vertices admits a $1$-factorization, showing that $\lambda \leq cd$ is sufficient for some absolute constant $c > 0$.