Dynamic Algorithms for Graph Coloring

We design fast dynamic algorithms for proper vertex and edge colorings in a graph undergoing edge insertions and deletions. In the static setting, there are simple linear time algorithms for (∆ + 1)vertex coloring and (2∆ − 1)-edge coloring in a graph with maximum degree ∆. It is natural to ask if we can efficiently maintain such colorings in the dynamic setting as well. We get the following three results. (1) We present a randomized algorithm which maintains a (∆ + 1)-vertex coloring with O(log ∆) expected amortized update time. (2) We present a deterministic algorithm which maintains a (1 + o(1))∆-vertex coloring with O(polylog ∆) amortized update time. (3) We present a simple, deterministic algorithm which maintains a (2∆ − 1)-edge coloring with O(log ∆) worst-case update time. This improves the recent O(∆)-edge coloring algorithm with Õ( √ ∆) worst-case update time [BM17]. ∗Corresponding author. University of Warwick, UK. Email: [email protected] †Dartmouth College, USA. Email: [email protected] ‡University of Vienna, Austria. Email: [email protected] §KTH, Sweden. Email: [email protected]