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

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