  We present an $O(\log^3\log n)$-round distributed algorithm for the
$(\Delta+1)$-coloring problem, where each node broadcasts only one $O(\log
n)$-bit message per round to its neighbors. Previously, the best such
broadcast-based algorithm required $O(\log n)$ rounds. If $\Delta \in
\Omega(\log^{3} n)$, our algorithm runs in $O(\log^* n)$ rounds. Our
algorithm's round complexity matches state-of-the-art in the much more powerful
CONGEST model [Halld\'orsson et al., STOC'21 & PODC'22], where each node sends
one different message to each of its neighbors, thus sending up to
$\Theta(n\log n)$ bits per round. This is the best complexity known, even if
message sizes are unbounded.
  Our algorithm is simple enough to be implemented in even weaker models: we
can achieve the same $O(\log^3\log n)$ round complexity if each node reads its
received messages in a streaming fashion, using only $O(\log^3 n)$-bit memory.
Therefore, we hope that our algorithm opens the road for adopting the recent
exciting progress on sublogarithmic-time distributed $(\Delta+1)$-coloring
algorithms in a wider range of (theoretical or practical) settings.
