The critical Karp--Sipser core of Erdős--Rényi random graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913599260196864 |
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| author | Budzinski, Thomas Contat, Alice |
| author_facet | Budzinski, Thomas Contat, Alice |
| contents | 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, λ/n]$, it is known to exhibit a phase transition at $λ= \mathrm{e}$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_04328 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The critical Karp--Sipser core of Erdős--Rényi random graphs Budzinski, Thomas Contat, Alice Probability Combinatorics 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, λ/n]$, it is known to exhibit a phase transition at $λ= \mathrm{e}$. 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. |
| title | The critical Karp--Sipser core of Erdős--Rényi random graphs |
| topic | Probability Combinatorics |
| url | https://arxiv.org/abs/2412.04328 |