  We consider the problem of designing deterministic graph algorithms for the
model of Massively Parallel Computation (MPC) that improve with the sparsity of
the input graph, as measured by the notion of arboricity. For the problems of
maximal independent set (MIS), maximal matching (MM), and vertex coloring, we
improve the state of the art as follows. Let $\lambda$ denote the arboricity of
the $n$-node input graph with maximum degree $\Delta$.
  MIS and MM: We develop a deterministic low-space MPC algorithm that reduces
the maximum degree to $poly(\lambda)$ in $O(\log \log n)$ rounds, improving and
simplifying the randomized $O(\log \log n)$-round $poly(\max(\lambda, \log
n))$-degree reduction of Ghaffari, Grunau, Jin [DISC'20]. Our approach when
combined with the state-of-the-art $O(\log \Delta + \log \log n)$-round
algorithm by Czumaj, Davies, Parter [SPAA'20, TALG'21] leads to an improved
deterministic round complexity of $O(\log \lambda + \log \log n)$ for MIS and
MM in low-space MPC.
  We also extend above MIS and MM algorithms to work with linear global memory.
Specifically, we show that both problems can be solved in deterministic time
$O(\min(\log n, \log \lambda \cdot \log \log n))$, and even in $O(\log \log n)$
time for graphs with arboricity at most $\log^{O(1)} \log n$. In this setting,
only a $O(\log^2 \log n)$-running time bound for trees was known due to Latypov
and Uitto [ArXiv'21].
  Vertex Coloring: We present a $O(1)$-round deterministic algorithm for the
problem of $O(\lambda)$-coloring in linear-memory MPC with relaxed global
memory of $n \cdot poly(\lambda)$ that solves the problem after just one single
graph partitioning step. This matches the state-of-the-art randomized round
complexity by Ghaffari and Sayyadi [ICALP'19] and improves upon the
deterministic $O(\lambda^{\epsilon})$-round algorithm by Barenboim and Khazanov
[CSR'18].
