Explicit and Implicit Dynamic Coloring of Graphs with Bounded Arboricity

Abstract

Graph coloring is a fundamental problem in computer science. We study the fully dynamic version of the problem in which the graph is undergoing edge insertions and deletions and we wish to maintain a vertex-coloring with small update time after each insertion and deletion. We show how to maintain an $O(\alpha \lg n)$-coloring with polylogarithmic update time, where $n$ is the number of vertices in the graph and $\alpha$ is the current arboricity of the graph. This improves upon a result by Solomon and Wein (ESA'18) who maintained an $O(\alpha_{\max}\lg^2 n)$-coloring, where $\alpha_{\max}$ is the maximum arboricity of the graph over all updates. Furthermore, motivated by a lower bound by Barba et al. (Algorithmica'19), we initiate the study of implicit dynamic colorings. Barba et al. showed that dynamic algorithms with polylogarithmic update time cannot maintain an $f(\alpha)$-coloring for any function $f$ when the vertex colors are stored explicitly, i.e., for each vertex the color is stored explicitly in the memory. Previously, all dynamic algorithms maintained explicit colorings. Therefore, we propose to study implicit colorings, i.e., the data structure only needs to offer an efficient query procedure to return the color of a vertex (instead of storing its color explicitly). We provide an algorithm which breaks the lower bound and maintains an implicit $2^{O(\alpha)}$-coloring with polylogarithmic update time. In particular, this yields the first dynamic $O(1)$-coloring for graphs with constant arboricity such as planar graphs or graphs with bounded tree-width, which is impossible using explicit colorings. We also show how to dynamically maintain a partition of the graph's edges into $O(\alpha)$ forests with polylogarithmic update time. We believe this data structure is of independent interest and might have more applications in the future.