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Monotonicity and decompositions of random regular graphs

Lawrence Hollom, Lyuben Lichev, Adva Mond, Julien Portier, Yiting Wang

TL;DR

This paper addresses how random regular graphs of different degrees relate to one another, focusing on monotone couplings and decompositions analogous to sprinkling in Erdős–Rényi graphs. It introduces two key tools: (i) contiguity and total variation bounds under edge-disjoint unions (the ⊕ operation) and (ii) a novel Alternative Sampling Procedure (ASP) for generating unions of disjoint perfect matchings, enabling precise approximations of $G(n,d)$ by sums $G(n,d_1)oxplus G(n,d_2-d_1)$ and by repeated $G(n,1)$ factors. The authors prove that, in a new regime (for even $n$ and $d_1 o n^{1/10}$, among other ranges), there exists a coupling with high probability such that $G(n,d_1) subseteq G(n,d_2)$, and they show contiguity between related decomposed models, yielding total variation bounds of the form $d_{ ext{TV}}(oldsymbol{ u}_doxplusoldsymbol{ u}_1,oldsymbol{ u}_{d+1}) o 0$. A central technical contribution is a refined analysis of the number of perfect matchings and the triangle count via orthogonal decomposition and moment calculations up to the 4th moment, which sharpens Gao’s bounds and underpins the contiguity arguments. These results advance understanding of monotonicity, contiguity, and factorisations in random regular graphs and open avenues toward resolving related conjectures by Isaev, McKay, Southwell, and Zhukovskii.

Abstract

In this work we establish several monotonicity and decomposition results in the framework of random regular graphs. Among other results, we show that, for a wide range of parameters $d_1 \leq d_2$, there exists a coupling of $G(n,d_1)$ and $G(n,d_2)$ satisfying that $G(n,d_1) \subseteq G(n,d_2)$ with high probability, confirming a conjecture of Gao, Isaev and McKay in a new regime. Our contributions include new tools for analysing contiguity and total variation distance between random regular graph models, a novel procedure for generating unions of random edge-disjoint perfect matchings, and refined estimates of Gao's bounds on the number of perfect matchings in random regular graphs. In addition, we make progress towards another conjecture of Isaev, McKay, Southwell and Zhukovskii.

Monotonicity and decompositions of random regular graphs

TL;DR

This paper addresses how random regular graphs of different degrees relate to one another, focusing on monotone couplings and decompositions analogous to sprinkling in Erdős–Rényi graphs. It introduces two key tools: (i) contiguity and total variation bounds under edge-disjoint unions (the ⊕ operation) and (ii) a novel Alternative Sampling Procedure (ASP) for generating unions of disjoint perfect matchings, enabling precise approximations of by sums and by repeated factors. The authors prove that, in a new regime (for even and , among other ranges), there exists a coupling with high probability such that , and they show contiguity between related decomposed models, yielding total variation bounds of the form . A central technical contribution is a refined analysis of the number of perfect matchings and the triangle count via orthogonal decomposition and moment calculations up to the 4th moment, which sharpens Gao’s bounds and underpins the contiguity arguments. These results advance understanding of monotonicity, contiguity, and factorisations in random regular graphs and open avenues toward resolving related conjectures by Isaev, McKay, Southwell, and Zhukovskii.

Abstract

In this work we establish several monotonicity and decomposition results in the framework of random regular graphs. Among other results, we show that, for a wide range of parameters , there exists a coupling of and satisfying that with high probability, confirming a conjecture of Gao, Isaev and McKay in a new regime. Our contributions include new tools for analysing contiguity and total variation distance between random regular graph models, a novel procedure for generating unions of random edge-disjoint perfect matchings, and refined estimates of Gao's bounds on the number of perfect matchings in random regular graphs. In addition, we make progress towards another conjecture of Isaev, McKay, Southwell and Zhukovskii.

Paper Structure

This paper contains 18 sections, 21 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Outline of the main ideas.
  3. Organisation of the paper.
  4. Notation and preliminaries
  5. Notation and terminology
  6. General notation.
  7. Coupling.
  8. Total variation distance.
  9. Contiguity.
  10. Preliminary results
  11. Helpful results from the literature
  12. Other preliminary results.
  13. Main technical result: decomposing one matching at a time
  14. Proofs of Theorem \ref{['thm:Coupling-Sprinkling-2-Random']} and \ref{['thm:Contiguity-Disjoint-Matchings']}
  15. Coupling lemmas: prelude to the proof of Theorem \ref{['thm:Coupling-Inclusion']}
  16. ...and 3 more sections

Key Result

Theorem 1.2

Consider $n$ even, $d_1\in [3,n^{1/7}/\log n]$ and $d_2\in [d_1,n-1]$ with $d_2 = \omega(1)$. Then there exists a coupling of $G_1 \sim G(n,d_1)$ and $G_{2} \sim G(n,d_2)$ such that $\mathbb{P}(G_1 \subseteq G_2)=1-o(1)$.

Theorems & Definitions (48)

  • Conjecture 1.1: Conjecture 1.2 in GIM22
  • Theorem 1.2
  • Conjecture 1.3: Conjecture 1.2 in IMcKSZ23
  • Theorem 1.4
  • Theorem 1.5
  • Conjecture 1.6
  • Theorem 2.1: Theorem 4.6 in McK85
  • Theorem 2.2: see Corollary 4.17 in Wor99
  • Theorem 2.3: Corollary 2.2 from IMcKSZ23
  • Lemma 2.4: see Theorem 2.12 in Den12 and its proof
  • ...and 38 more