The critical Karp--Sipser core of Erdős--Rényi random graphs

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 Erdős–Rényi random graph $\mathrm{G}[n,\lambda \mathrm{/}n]$, 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 $n^{3\mathrm{/}5}$, 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.