The typical structure of dense claw-free graphs
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916589136248832 |
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| author | Perkins, Will van der Poel, Sam |
| author_facet | Perkins, Will van der Poel, Sam |
| contents | We analyze the asymptotic number and typical structure of claw-free graphs at constant edge densities. The first of our main results is a formula for the asymptotics of the logarithm of the number of claw-free graphs of edge density $γ\in (0,1)$. We show that the problem exhibits a second-order phase transition at edge density $γ^\ast=\frac{5-\sqrt{5}}{4}$. The asymptotic formula arises by solving a variational problem over graphons. For $γ\geqγ^\ast$ there is a unique optimal graphon, while for $γ<γ^\ast$ there is an infinite set of optimal graphons. By analyzing more detailed structure, we prove that for $γ<γ^\ast$, there is in fact a unique graphon $W$ such that almost all claw-free graphs at edge density $γ$ are close in cut metric to $W$. We also analyze the probability of claw-freeness in the Erdős-Rényi random graph $G(n,p)$ for constant $p$, obtaining a formula for the large-deviation rate function for claw-freeness. In this case, the problem exhibits a first-order phase transition at $p^\ast=\frac{3-\sqrt{5}}{2}$, separating distinct structural regimes. At the critical point $p^\ast$, the corresponding graphon variational problem has infinitely many solutions, and we again pinpoint a unique optimal graphon that describes the typical structure of $G(n,p^\ast)$ conditioned on being claw-free. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_17816 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The typical structure of dense claw-free graphs Perkins, Will van der Poel, Sam Combinatorics Probability We analyze the asymptotic number and typical structure of claw-free graphs at constant edge densities. The first of our main results is a formula for the asymptotics of the logarithm of the number of claw-free graphs of edge density $γ\in (0,1)$. We show that the problem exhibits a second-order phase transition at edge density $γ^\ast=\frac{5-\sqrt{5}}{4}$. The asymptotic formula arises by solving a variational problem over graphons. For $γ\geqγ^\ast$ there is a unique optimal graphon, while for $γ<γ^\ast$ there is an infinite set of optimal graphons. By analyzing more detailed structure, we prove that for $γ<γ^\ast$, there is in fact a unique graphon $W$ such that almost all claw-free graphs at edge density $γ$ are close in cut metric to $W$. We also analyze the probability of claw-freeness in the Erdős-Rényi random graph $G(n,p)$ for constant $p$, obtaining a formula for the large-deviation rate function for claw-freeness. In this case, the problem exhibits a first-order phase transition at $p^\ast=\frac{3-\sqrt{5}}{2}$, separating distinct structural regimes. At the critical point $p^\ast$, the corresponding graphon variational problem has infinitely many solutions, and we again pinpoint a unique optimal graphon that describes the typical structure of $G(n,p^\ast)$ conditioned on being claw-free. |
| title | The typical structure of dense claw-free graphs |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/2501.17816 |