  Given a graph $G$ and $p \in [0,1]$, let $G_p$ denote the random subgraph of
$G$ obtained by keeping each edge independently with probability $p$. Alon,
Krivelevich, and Sudokov proved $\mathbb{E} [\chi(G_p)] \geq C_p
\frac{\chi(G)}{\log |V(G)|}$, and Bukh conjectured an improvement of
$\mathbb{E}[\chi(G_p)] \geq C_p \frac{\chi(G)}{\log \chi(G)}$. We prove a new
spectral lower bound on $\mathbb{E}[\chi(G_p)]$, as progress towards Bukh's
conjecture. We also propose the stronger conjecture that for any fixed $p \leq
1/2$, among all graphs of fixed chromatic number, $\mathbb{E}[\chi(G_p)]$ is
minimized by the complete graph. We prove this stronger conjecture when $G$ is
planar or $\chi(G) < 4$. We also consider weaker lower bounds on
$\mathbb{E}[\chi(G_p)]$ proposed in a recent paper by Shinkar; we answer two
open questions of Shinkar negatively and propose a possible refinement of one
of them.
