  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 $G_{1/2}$ is at most $k/3 + o(k)$ with high probability, thus disproving the
natural prediction that such random graphs must have colouring number at least
$k/2 - o(k)$.
