Colouring random subgraphs

We study several basic problems about colouring the $p$-random subgraph $G_p$ of an arbitrary graph $G$, focusing primarily on the chromatic number and colouring number of $G_p$. In particular, we show that there exist infinitely many $k$-regular graphs $G$ for which the colouring number (i.e., degeneracy) of $G_{1\mathrm{/}2}$ is at most $k\mathrm{/}3+o(k)$ with high probability, thus disproving the natural prediction that such random graphs must have colouring number at least $k\mathrm{/}2-o(k)$.