  In this paper, we present improved algorithms for the $(\Delta+1)$ (vertex)
coloring problem in the Congested-Clique model of distributed computing. In
this model, the input is a graph on $n$ nodes, initially each node knows only
its incident edges, and per round each two nodes can exchange $O(\log n)$ bits
of information.
  Our key result is a randomized $(\Delta+1)$ vertex coloring algorithm that
works in $O(\log\log \Delta \cdot \log^* \Delta)$-rounds. This is achieved by
combining the recent breakthrough result of [Chang-Li-Pettie, STOC'18] in the
\local\ model and a degree reduction technique. We also get the following
results with high probability: (1) $(\Delta+1)$-coloring for $\Delta=O((n/\log
n)^{1-\epsilon})$ for any $\epsilon \in (0,1)$, within
$O(\log(1/\epsilon)\log^* \Delta)$ rounds, and (2)
$(\Delta+\Delta^{1/2+o(1)})$-coloring within $O(\log^* \Delta)$ rounds. Turning
to deterministic algorithms, we show a $(\Delta+1)$-coloring algorithm that
works in $O(\log \Delta)$ rounds.
