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(n,p) when p=(1+& varepsilon;)/(n) for & varepsilon;>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 & varepsilon;. Finally, we also consider the width of the random graph in the weakly supercritical regime, where & varepsilon;=o(1) and & varepsilon;(3)n ->infinity. 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 & varepsilon;.