  In recent years, there has been a growing interest in solving various graph
coloring problems in the streaming model. The initial algorithms in this line
of work are all crucially randomized, raising natural questions about how
important a role randomization plays in streaming graph coloring. A couple of
very recent works have made progress on this question: they prove that
deterministic or even adversarially robust coloring algorithms (that work on
streams whose updates may depend on the algorithm's past outputs) are
considerably weaker than standard randomized ones. However, there is still a
significant gap between the upper and lower bounds for the number of colors
needed (as a function of the maximum degree $\Delta$) for robust coloring and
multipass deterministic coloring. We contribute to this line of work by proving
the following results.
  In the deterministic semi-streaming (i.e., $O(n \cdot \text{polylog } n)$
space) regime, we present an algorithm that achieves a combinatorially optimal
$(\Delta+1)$-coloring using $O(\log{\Delta} \log\log{\Delta})$ passes. This
improves upon the prior $O(\Delta)$-coloring algorithm of Assadi, Chen, and Sun
(STOC 2022) at the cost of only an $O(\log\log{\Delta})$ factor in the number
of passes.
  In the adversarially robust semi-streaming regime, we design an
$O(\Delta^{5/2})$-coloring algorithm that improves upon the previously best
$O(\Delta^{3})$-coloring algorithm of Chakrabarti, Ghosh, and Stoeckl (ITCS
2022). Further, we obtain a smooth colors/space tradeoff that improves upon
another algorithm of the said work: whereas their algorithm uses $O(\Delta^2)$
colors and $O(n\Delta^{1/2})$ space, ours, in particular, achieves
(i)~$O(\Delta^2)$ colors in $O(n\Delta^{1/3})$ space, and
(ii)~$O(\Delta^{7/4})$ colors in $O(n\Delta^{1/2})$ space.
