The problem of (vertex) $(Delta+1)$-coloring a graph of maximum degree $Delta$ has been extremely well-studied over the years in various settings and models. Surprisingly, for the dynamic setting, almost nothing was known until recently. In SODA18, Bhattacharya, Chakrabarty, Henzinger and Nanongkai devised a randomized data structure for maintaining a $(Delta+1)$-coloring with $O(log Delta)$ expected amortized update time. In this paper, we present a $(Delta+1)$-coloring data structure that achieves a constant amortized update time and show that this time bound holds not only in expectation but also with high probability.
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