  The main results of this paper are (I) a simulation algorithm which, under
quite general constraints, transforms algorithms running on the Congested
Clique into algorithms running in the MapReduce model, and (II) a distributed
$O(\Delta)$-coloring algorithm running on the Congested Clique which has an
expected running time of (i) $O(1)$ rounds, if $\Delta \geq \Theta(\log^4 n)$;
and (ii) $O(\log \log n)$ rounds otherwise. Applying the simulation theorem to
the Congested-Clique $O(\Delta)$-coloring algorithm yields an $O(1)$-round
$O(\Delta)$-coloring algorithm in the MapReduce model.
  Our simulation algorithm illustrates a natural correspondence between
per-node bandwidth in the Congested Clique model and memory per machine in the
MapReduce model. In the Congested Clique (and more generally, any network in
the $\mathcal{CONGEST}$ model), the major impediment to constructing fast
algorithms is the $O(\log n)$ restriction on message sizes. Similarly, in the
MapReduce model, the combined restrictions on memory per machine and total
system memory have a dominant effect on algorithm design. In showing a fairly
general simulation algorithm, we highlight the similarities and differences
between these models.
