The critical Karp--Sipser core of Erd\H{o}s--R\'enyi random graphs

TL;DR

This paper analyzes the size and distribution of the Karp--Sipser core in Erdős–Rényi random graphs at the phase transition point, confirming a conjecture and providing detailed asymptotic results.

Contribution

It proves the size of the Karp--Sipser core at criticality is of order n^{3/5} and describes its asymptotic distribution, confirming a conjecture of Bauer and Golinelli.

Findings

Karp--Sipser core size at criticality is of order n^{3/5}

Asymptotic law of the core's size is established

Distribution of the core as a graph is characterized

Abstract

The Karp--Sipser algorithm consists in removing recursively the leaves as well their unique neighbours and all isolated vertices of a given graph. The remaining graph obtained when there is no leaf left is called the Karp--Sipser core. When the underlying graph is the classical sparse Erd\H{o}s--R\'enyi random graph $ \mathrm{G}[n, \lambda/n]$, it is known to exhibit a phase transition at $\lambda = \mathrm{e}$. We show that at criticality, the Karp--Sipser core has size of order $n^{3/5}$, which proves a conjecture of Bauer and Golinelli. We provide the asymptotic law of this renormalized size as well as a description of the distribution of the core as a graph. Our approach relies on the differential equation method, and builds up on a previous work on a configuration model with bounded degrees.