A note on the chromatic number of the square of a sparse random graph

We show that w.h.p the list chromatic number $\chi_\ell$ of the square of $G_{n,p}$ for $p=c/n$ is asymptotically equal to the maximum degree $\Delta(G_{n,p})$. Since $\chi(G^2_{n,p})\leq \chi_\ell(G^2_{n,p})$, this also improves an earlier result of Garapaty et al \cite{KLMP} who proved that $\chi(G^2_{n,p}) \leq 6 \cdot \Delta(G_{n,p})$ w.h.p.