A note on the width of sparse random graphs

TL;DR

This paper provides simplified proofs and explicit bounds for the rank- and tree-width of supercritical random graphs, including the weakly supercritical regime, improving understanding of their structural properties.

Contribution

It offers direct proofs avoiding complex expansion results and determines the width of random graphs in the weakly supercritical regime with explicit bounds.

Findings

Explicit bounds on width depending on epsilon

Simplified proofs of known results

Width characterization in weakly supercritical regime

Abstract

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 $p= \frac{1+\epsilon}{n}$ for $\epsilon > 0$ 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 $\epsilon$. Finally, we also consider the width of the random graph in the weakly supercritical regime, where $\epsilon = o(1)$ and $\epsilon^3n \to \infty$. 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 $\epsilon$.