  Given a system $(G_1, \ldots ,G_m)$ of graphs on the same vertex set $V$, a
cooperative coloring is a choice of vertex sets $I_1, \ldots ,I_m$, such that
$I_j$ is independent in $G_j$ and $\bigcup_{j=1}^{m}I_j = V$. For a class
$\mathcal{G}$ of graphs, let $m_{\mathcal{G}}(d)$ be the minimal $m$ such that
every $m$ graphs from $\mathcal{G}$ with maximum degree $d$ have a cooperative
coloring. We prove that $\Omega(\log\log d) \le m_\mathcal{T}(d) \le O(\log d)$
and $\Omega(\log d)\le m_\mathcal{B}(d) \le O(d/\log d)$, where $\mathcal{T}$
is the class of trees and $\mathcal{B}$ is the class of bipartite graphs.
