Coloring powers of random graphs

Abstract

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 $\Delta(G_{n,p}^r)$ and the chromatic number $\chi(G_{n,p}^r)$, 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.~$\Delta(G_{n,p}^r) \sim \frac{\log n}{\log_{(r+1)}n}$ (where $\log_{(1)}n=\log n$ and $\log_{(r+1)}n=\log\log_{(r)}n$) and that w.h.p.~$\Delta(G_{n,p}^{\lfloor{r/2}\rfloor})+1 \le \chi(G_{n,p}^r) \le \Delta(G_{n,p}^{r-1})+1$. For $r=2$, we show the upper bound holds with equality. For denser cases, for $d$ satisfying $d=\omega(\log n)$ and $d\le n^{1/r-\Omega(1)}$ as $n\to\infty$, we have $\chi(G_{n,p}^r) = \Theta(d^r/\log d)$ w.h.p.