Dynamic coloring for Bipartite and General Graphs
Abstract
We consider the dynamic coloring problem on bipartite and general graphs in the incremental as well as fullydynamic 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 tradeoff 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 fullydynamic 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2019
 arXiv:
 arXiv:1909.07854
 Bibcode:
 2019arXiv190907854J
 Keywords:

 Computer Science  Data Structures and Algorithms