Coloring powers of random graphs
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866913035489116160 |
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| author | Frieze, Alan Kang, Ross Raut, Aditya Sweering, Michelle Verbeek, Hilde |
| author_facet | Frieze, Alan Kang, Ross Raut, Aditya Sweering, Michelle Verbeek, Hilde |
| contents | Given a graph $G$ and an integer $r\ge 1$, the $r$th power $G^r$ of $G$ is the graph obtained from $G$ by adding edges for all pairs of distinct vertices at distance at most $r$ from each other. We focus on two basic structural properties of the $r$th power of the binomial random graph $G_{n,p}$, namely, the maximum degree $Δ(G_{n,p}^r)$ and the chromatic number $χ(G_{n,p}^r)$, and give with high probability (w.h.p.) bounds.
In the sparse case that $p=d/n$ for some fixed constant $d>0$, we prove the following. We prove that w.h.p.~$Δ(G_{n,p}^r) \sim \frac{\log n}{\log_{(r+1)}n}$ (where $\log_{(1)}n=\log n$ and $\log_{(r+1)}n=\log\log_{(r)}n$) and that w.h.p.~$Δ(G_{n,p}^{\lfloor{r/2}\rfloor})+1 \le χ(G_{n,p}^r) \le Δ(G_{n,p}^{r-1})+1$. For $r=2$, we show the upper bound holds with equality.
For denser cases, for $d$ satisfying $d=ω(\log n)$ and $d\le n^{1/r-Ω(1)}$ as $n\to\infty$, we have $χ(G_{n,p}^r) = Θ(d^r/\log d)$ w.h.p. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_14006 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Coloring powers of random graphs Frieze, Alan Kang, Ross Raut, Aditya Sweering, Michelle Verbeek, Hilde Combinatorics Given a graph $G$ and an integer $r\ge 1$, the $r$th power $G^r$ of $G$ is the graph obtained from $G$ by adding edges for all pairs of distinct vertices at distance at most $r$ from each other. We focus on two basic structural properties of the $r$th power of the binomial random graph $G_{n,p}$, namely, the maximum degree $Δ(G_{n,p}^r)$ and the chromatic number $χ(G_{n,p}^r)$, and give with high probability (w.h.p.) bounds. In the sparse case that $p=d/n$ for some fixed constant $d>0$, we prove the following. We prove that w.h.p.~$Δ(G_{n,p}^r) \sim \frac{\log n}{\log_{(r+1)}n}$ (where $\log_{(1)}n=\log n$ and $\log_{(r+1)}n=\log\log_{(r)}n$) and that w.h.p.~$Δ(G_{n,p}^{\lfloor{r/2}\rfloor})+1 \le χ(G_{n,p}^r) \le Δ(G_{n,p}^{r-1})+1$. For $r=2$, we show the upper bound holds with equality. For denser cases, for $d$ satisfying $d=ω(\log n)$ and $d\le n^{1/r-Ω(1)}$ as $n\to\infty$, we have $χ(G_{n,p}^r) = Θ(d^r/\log d)$ w.h.p. |
| title | Coloring powers of random graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.14006 |