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Sandwiching between random regular graphs and Erdős-Rényi graphs: configuration model and unions of perfect matchings

Pu Gao, Mikhail Isaev, Xavier Perez-Gimenez

Abstract

We establish new couplings among several random graph and multigraph models related to the random regular graph $G(n,d)$, including the configuration model and unions of random perfect matchings. As a main result, we verify the Kim-Vusandwich conjecture for all large degrees $d=n-O(\log^4 n)$ and prove a weakened version for $d=O(\log^4 n)$, which are the only remaining open cases. Our approach introduces a coupling framework that links $G(n,d)$ and $G(n,p)$ through a chain of intermediate models.

Sandwiching between random regular graphs and Erdős-Rényi graphs: configuration model and unions of perfect matchings

Abstract

We establish new couplings among several random graph and multigraph models related to the random regular graph , including the configuration model and unions of random perfect matchings. As a main result, we verify the Kim-Vusandwich conjecture for all large degrees and prove a weakened version for , which are the only remaining open cases. Our approach introduces a coupling framework that links and through a chain of intermediate models.
Paper Structure (20 sections, 33 theorems, 125 equations)

This paper contains 20 sections, 33 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Bridging by unions of perfect matchings
  3. Decomposition of the loopless configuration model
  4. Embedding ${\mathcal{G}}(n,d)$ into $\mathcal{P}_*(n,d+1)$
  5. Tools
  6. Small $d$: proof of Theorem \ref{['thm:smalld']}
  7. Proof of Theorem \ref{['thm:P-P']}
  8. Concentration of the number of perfect matchings
  9. Proof of Theorem \ref{['thm:P-G']}
  10. Concentration of $Y_n^{(i)}$ in $\mathcal{P}^{(i)}(n,d)$
  11. Proof of Theorem \ref{['thm:P-G']}
  12. Proofs of Theorems \ref{['thm:P-P2']} and \ref{['thm:P-M']} and Corollary \ref{['thm:G-M']}
  13. Concentration of the number of perfect matchings
  14. Proof of Theorem \ref{['thm:P-P2']}
  15. Proof of Theorem \ref{['thm:P-M']}
  16. ...and 5 more sections

Key Result

Proposition 1.1

${\mathcal{G}}(n,d)$ is equivalent to $G[P]$ for $P\sim \mathcal{P}(n,d)$ conditional on the event that $G[P]$ is a simple graph.

Theorems & Definitions (39)

  • Proposition 1.1: wormald1999models
  • Proposition 1.2: wormald1999models
  • Proposition 1.3: wormald1999models
  • Proposition 1.4: gao2020kim
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 29 more