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Regular bipartite decompositions of pseudorandom graphs

Asaf Ferber, Bryce Frederickson, Dingjia Mao, Liana Yepremyan, Yizhe Zhu

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 $(n,d,λ)$-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 $(n,d,λ)$-graph with $λ\leq d/12$ 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 $λ= λ(d)$ by Ferber and Jain (2020) to guarantee that an $(n,d,λ)$-graph on an even number of vertices admits a $1$-factorization, showing that $λ\leq cd$ is sufficient for some absolute constant $c > 0$.

Regular bipartite decompositions of pseudorandom graphs

Abstract

In 1972, Kotzig proved that for every even , the complete graph can be decomposed into edge-disjoint regular bipartite spanning subgraphs, which is best possible. In this paper, we study regular bipartite decompositions of -graphs, where is an even integer and for some absolute constant . With a randomized algorithm, we prove that such an -graph with can be decomposed into at most 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 by Ferber and Jain (2020) to guarantee that an -graph on an even number of vertices admits a -factorization, showing that is sufficient for some absolute constant .

Paper Structure

This paper contains 11 sections, 17 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Proof outline
  4. Preliminaries
  5. Probabilistic tools
  6. Tools in graph theory
  7. Almost regular decomposition via random bisections
  8. Regularization
  9. Pairing subgraphs with the same number of edges
  10. Obtaining a regular decomposition from almost regular decompositions
  11. Proof of our main result

Key Result

Theorem 1.1

Let $n \geq 2$ be an even integer. Then the edge set of the complete graph $K_n$ can be decomposed into $\lceil\log_2 n\rceil$ regular bipartite spanning subgraphs, which is best possible.

Figures (1)

  • Figure 1: Blue edges represent edges of $H_X$ and $H_Y$, and red edges represent edges in $R'$ and $R"$. $H_X\cup X_Y\cup R' \cup R"$ is a bipartite regular graph with bipartition $\{X'\cup Y", X"\cup Y'\}$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2: Theorem 1.1 in ferber20181
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 2.1: Chernoff's bounds
  • Remark 2.2
  • Lemma 2.3: Lovász Local Lemma (asymmetric version)
  • Lemma 2.4: Lovász Local Lemma (symmetric version)
  • Lemma 2.5: Expander mixing lemma
  • Theorem 2.6: Vizing's theorem
  • ...and 22 more