Fully Dynamic $(\Delta +1)$-Coloring in Constant Update Time

Abstract

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 SODA'18, 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.