1 research outputs found
Explicit and Implicit Dynamic Coloring of Graphs with Bounded Arboricity
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 -coloring with polylogarithmic
update time, where is the number of vertices in the graph and is
the current arboricity of the graph. This improves upon a result by Solomon and
Wein (ESA'18) who maintained an -coloring, where
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
-coloring for any function 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 -coloring with polylogarithmic
update time. In particular, this yields the first dynamic -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 forests with polylogarithmic update time. We believe this data
structure is of independent interest and might have more applications in the
future