Coloring powers of random graphs

TL;DR

This paper investigates the structural properties of the rth power of random graphs, providing bounds on maximum degree and chromatic number in sparse and denser regimes with high probability.

Contribution

It establishes new probabilistic bounds on maximum degree and chromatic number of the rth power of G(n,p), extending understanding in both sparse and denser graph regimes.

Findings

Maximum degree of G_{n,p}^r is approximately log n / log_{(r+1)} n w.h.p.

Chromatic number bounds relate to maximum degrees, with exact bounds for r=2.

Chromatic number scales as Theta(d^r / log d) in denser regimes.

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.