A maximal independent set and a (deg+1)-coloring of any graph can be computed deterministically in O(n+m) work and polylog depth, matching the sequential greedy bound.
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Truly Work-efficient Parallel Deterministic $(\Delta+1)$-coloring and Maximal Independent Set
A maximal independent set and a (deg+1)-coloring of any graph can be computed deterministically in O(n+m) work and polylog depth, matching the sequential greedy bound.