The chromatic number of very dense random graphs
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909209804668928 |
|---|---|
| author | Yan, Zhifei |
| author_facet | Yan, Zhifei |
| contents | The chromatic number of a very dense random graph $G(n,p)$, with $p \ge 1 - n^{-c}$ for some constant $c > 0$, was first studied by Surya and Warnke, who conjectured that the typical deviation of $χ(G(n,p))$ from its mean is of order $\sqrt{μ_r}$, where $μ_r$ is the expected number of independent sets of size $r$, and $r$ is maximal such that $μ_r > 1$, except when $μ_r = O(\log n)$. They moreover proved their conjecture in the case $n^{-2} \ll 1 - p = O(n^{-1})$.
In this paper, we study $χ(G(n,p))$ in the range $n^{-1}\log n \ll 1 - p \ll n^{-2/3}$, that is, when the largest independent set of $G(n,p)$ is typically of size 3. We prove in this case that $χ(G(n,p))$ is concentrated on some interval of length $O(\sqrt{μ_3})$, and for sufficiently `smooth' functions $p = p(n)$, that there are infinitely many values of $n$ such that $χ(G(n,p))$ is not concentrated on any interval of size $o(\sqrt{μ_3})$. We also show that $χ(G(n,p))$ satisfies a central limit theorem in the range $n^{-1} \log n \ll 1 - p \ll n^{-7/9}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_13914 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The chromatic number of very dense random graphs Yan, Zhifei Combinatorics Probability The chromatic number of a very dense random graph $G(n,p)$, with $p \ge 1 - n^{-c}$ for some constant $c > 0$, was first studied by Surya and Warnke, who conjectured that the typical deviation of $χ(G(n,p))$ from its mean is of order $\sqrt{μ_r}$, where $μ_r$ is the expected number of independent sets of size $r$, and $r$ is maximal such that $μ_r > 1$, except when $μ_r = O(\log n)$. They moreover proved their conjecture in the case $n^{-2} \ll 1 - p = O(n^{-1})$. In this paper, we study $χ(G(n,p))$ in the range $n^{-1}\log n \ll 1 - p \ll n^{-2/3}$, that is, when the largest independent set of $G(n,p)$ is typically of size 3. We prove in this case that $χ(G(n,p))$ is concentrated on some interval of length $O(\sqrt{μ_3})$, and for sufficiently `smooth' functions $p = p(n)$, that there are infinitely many values of $n$ such that $χ(G(n,p))$ is not concentrated on any interval of size $o(\sqrt{μ_3})$. We also show that $χ(G(n,p))$ satisfies a central limit theorem in the range $n^{-1} \log n \ll 1 - p \ll n^{-7/9}$. |
| title | The chromatic number of very dense random graphs |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/2405.13914 |