Dynamic coloring for Bipartite and General Graphs

We consider the dynamic coloring problem on bipartite and general graphs in the incremental as well as fully-dynamic settings. In this work, we are interested in the following parameters : the update time and query time, the number of colors used, and the number of vertex recolorings per update. Our results reveal the following trade-off for a bipartite graph with $n$ vertices: In the fully dynamic setting, if we restrict the number of colors to $2$ then the maximum of update and query time is at least $\log n$. In the incremental setting, using $2$ colors we achieve the maximum of update and query time to be $O(\alpha(n))$, where $\alpha(n)$ is the inverse Ackermann function. We show that by allowing more than two colors we can reduce the query time to $O(1)$ without changing the update time. Our incremental algorithm uses $1+2 \log{n}$ colors. To the best of our knowledge, there are no known theoretical guarantees for dynamic coloring specific to bipartite graphs. For general graphs we provide a deterministic fully-dynamic algorithm with constant number of recolorings per update. We use $\Delta+1$ colors and achieve $O(\sqrt{m})$ worst case update time with at most one recoloring per update. Here $\Delta$ is the maximum degree of a vertex and $m$ denotes the maximum number of edges throughout the update sequence. For graphs of arboricity bounded by $\gamma$ we maintain a $\Delta+1$ coloring with at most one recoloring per update, an amortized update time of $O(\gamma + \log{n})$, and an $O(1)$ query time.