  We show an $\tilde{O}(n^{p/(p+2)})$-round algorithm in the \congest model for
\emph{listing} of $K_p$ (a clique with $p$ nodes), for all $p =4, p\geq 6$. For
$p = 5$, we show an $\tilde{O}(n^{3/4})$-round algorithm.
  For $p=4$ and $p=5$, our results improve upon the previous state-of-the-art
of $O(n^{5/6+o(1)})$ and $O(n^{21/22+o(1)})$, respectively, by Eden et al.
[DISC 2019]. For all $p\geq 6$, ours is the first sub-linear round algorithm
for $K_p$ listing.
  We leverage the recent expander decomposition algorithm of Chang et al. [SODA
2019] to create clusters with a good mixing time. Three key novelties in our
algorithm are: (1) we carefully iterate our listing process with coupled values
of min-degree within the clusters and arboricity outside the clusters, (2) all
the listing is done within the cluster, which necessitates new techniques for
bringing into the cluster the information about \emph{all} edges that can
potentially form $K_p$ instances with the cluster edges, and (3) within each
cluster we use a sparsity-aware listing algorithm, which is faster than a
general listing algorithm and which we can allow the cluster to use since we
make sure to sparsify the graph as the iterations proceed.
  As a byproduct of our algorithm, we show an \emph{optimal} sparsity-aware
algorithm for $K_p$ listing, which runs in $\tilde{\Theta}(1 + m/n^{1 + 2/p})$
rounds in the \clique model. Previously, Pandurangan et al. [SPAA 2018], Chang
et al. [SODA 2019], and Censor-Hillel et al. [TCS 2020] showed sparsity-aware
algorithms for the case of $p = 3$, yet ours is the first such sparsity aware
algorithm for $p \geq 4$.
