A note on the width of sparse random graphs

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 $G\left(n,p\right)$ when $p=\frac{1+ϵ}{n}$ for $ϵ>0$ constant. Our proofs avoid the use of black box results 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 $ϵ$. Finally, we also consider the width of the random graph in the weakly supercritical regime, where $ϵ=o\left(1\right)$ and ${ϵ}^{3}n\to \infty$. In this regime, we determine, up to a constant multiplicative factor, the rank- and tree-width of $G\left(n,p\right)$ as a function of $n$ and $ϵ$.