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The law of the circumference of sparse binomial random graphs

Michael Anastos, Joshua Erde, Mihyun Kang, Vincent Pfenninger

TL;DR

This work proves a central limit theorem for the circumference $L(G(n, rac{c}{n}))$ of sparse random graphs by developing a robust, multi-scale local approximation built around a strong $4$-core colouring. The authors decompose the circumference into a global quantity $L$ and a sequence of local approximations $ ilde{L}$, $ ilde{L}_k$, and $ ilde{L}_k^{\text{hat}}$, and transfer a CLT from the local neighbourhood counts to the global parameter via the Efron–Stein inequality and Stein’s method. A key contribution is a nontrivial variance lower bound achieved through a two-round edge-revealing scheme and the robust sapphire property, ensuring that fluctuations persist at scale $n$; this yields a limiting per-vertex variance $oldsymbol{ ho}^2$ with bounds $C_1 e^{-10c} \le \boldsymbol{ ho}^2 \le C_2 c$. The framework also offers a blueprint potentially applicable to other global graph parameters, such as the $k$-core and the matching number, illustrating a path to distributional results in sparse regimes beyond existing scaling laws.

Abstract

There has been much interest in the distribution of the circumference, the length of the longest cycle, of a random graph $G(n,p)$ in the sparse regime, when $p = Θ\left(\frac{1}{n}\right)$. Recently, the first author and Frieze established a scaling limit for the circumference in this regime, along the way establishing an alternative 'structural' approximation for this parameter. In this paper, we give a central limit theorem for the circumference in this regime using a novel argument based on the Efron-Stein inequality, which relies on a combinatorial analysis of the effect of resampling edges on this approximation.

The law of the circumference of sparse binomial random graphs

TL;DR

This work proves a central limit theorem for the circumference of sparse random graphs by developing a robust, multi-scale local approximation built around a strong -core colouring. The authors decompose the circumference into a global quantity and a sequence of local approximations , , and , and transfer a CLT from the local neighbourhood counts to the global parameter via the Efron–Stein inequality and Stein’s method. A key contribution is a nontrivial variance lower bound achieved through a two-round edge-revealing scheme and the robust sapphire property, ensuring that fluctuations persist at scale ; this yields a limiting per-vertex variance with bounds . The framework also offers a blueprint potentially applicable to other global graph parameters, such as the -core and the matching number, illustrating a path to distributional results in sparse regimes beyond existing scaling laws.

Abstract

There has been much interest in the distribution of the circumference, the length of the longest cycle, of a random graph in the sparse regime, when . Recently, the first author and Frieze established a scaling limit for the circumference in this regime, along the way establishing an alternative 'structural' approximation for this parameter. In this paper, we give a central limit theorem for the circumference in this regime using a novel argument based on the Efron-Stein inequality, which relies on a combinatorial analysis of the effect of resampling edges on this approximation.

Paper Structure

This paper contains 34 sections, 47 theorems, 283 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation and results
  3. Proof methods
  4. Main contributions of the paper
  5. Structure of the paper
  6. Proof of Theorem \ref{['thm:Law_of_longest_cycle']}
  7. The strong 4-core and a local approximation
  8. Approximation by neighbourhood counts
  9. Proof of Theorem \ref{['thm:Law_of_longest_cycle']}
  10. Notation and Preliminaries
  11. Probabilistic Preliminaries
  12. Stein's Method
  13. The effect of resampling edges on the global and local colourings
  14. Properties of the global and local colourings
  15. Localising the changes
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $c>1$ be a sufficiently large constant. Then there exists a function $f\colon (1,\infty)\to (0,\infty)$ such that

Figures (2)

  • Figure 1: The $3$-colouring of a graph $G$ given by \ref{['alg:strong_4_core_alg']}. Note that every vertex in $S\cup P$ has at least $4$ neighbours in $S$ and there is no edge between $R$ and $S$.
  • Figure 2: Vertices in $V\setminus (A \cup B)$ have the same neighbourhoods in $G_1$ and in $G_2$ but in $G_2$ more of these neighbours will be sapphire.

Theorems & Definitions (94)

  • Theorem 1.1: AF21
  • Theorem 1.2
  • Proposition 1.3
  • Theorem 2.1: A23, Theorem 11
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • Corollary 2.4
  • proof
  • Lemma 2.5
  • ...and 84 more