A note on the width of sparse random graphs
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911764989345792 |
|---|---|
| author | Do, Tuan Anh Erde, Joshua Kang, Mihyun |
| author_facet | Do, Tuan Anh Erde, Joshua Kang, Mihyun |
| contents | In this note, we consider the width of a supercritical random graph according to some commonly studied width measures. We give short, direct proofs of results of Lee, Lee and Oum, and of Perarnau and Serra, on the rank- and tree-width of the random graph $G(n,p)$ when $p= \frac{1+ε}{n}$ for $ε> 0$ constant. Our proofs avoid the use, as a black box, of a result of Benjamini, Kozma and Wormald on the expansion properties of the giant component in this regime, and so as a further benefit we obtain explicit bounds on the dependence of these results on $ε$. Finally, we also consider the width of the random graph in the weakly supercritical regime, where $ε= o(1)$ and $ε^3n \to \infty$. In this regime, we determine, up to a constant multiplicative factor, the rank- and tree-width of $G(n,p)$ as a function of $n$ and $ε$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_06087 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A note on the width of sparse random graphs Do, Tuan Anh Erde, Joshua Kang, Mihyun Combinatorics In this note, we consider the width of a supercritical random graph according to some commonly studied width measures. We give short, direct proofs of results of Lee, Lee and Oum, and of Perarnau and Serra, on the rank- and tree-width of the random graph $G(n,p)$ when $p= \frac{1+ε}{n}$ for $ε> 0$ constant. Our proofs avoid the use, as a black box, of a result of Benjamini, Kozma and Wormald on the expansion properties of the giant component in this regime, and so as a further benefit we obtain explicit bounds on the dependence of these results on $ε$. Finally, we also consider the width of the random graph in the weakly supercritical regime, where $ε= o(1)$ and $ε^3n \to \infty$. In this regime, we determine, up to a constant multiplicative factor, the rank- and tree-width of $G(n,p)$ as a function of $n$ and $ε$. |
| title | A note on the width of sparse random graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2202.06087 |