Weak graph colorings: distributed algorithms and applications

• Authors: Fabian Kuhn
• Venue: Proceedings of the twenty-first annual symposium on Parallelism in algorithms and architectures
• Year: 2009

We study deterministic, distributed algorithms for two weak variants of the standard graph coloring problem. We consider defective colorings, i.e., colorings where nodes of a color class may induce a graph of maximum degree d for some parameter d0. We also look at colorings where a minimum number of multi-chromatic edges is required. For an integer k0, we call a coloring k-partially proper if every node v has at least min{k,deg(v)} neighbors with a different color. We show that for all d∈{1,...,Δ}, it is possible to compute a O(Δ2/d2)-coloring with defect d in time O(log*n) where Δ is the largest degree of the network graph. Similarly, for all k∈{1,...,Δ}, a k-partially proper O(k2)-coloring can be computed in O(log*n) rounds. As an application of our weak defective coloring algorithm, we obtain a faster deterministic algorithm for the standard vertex coloring problem on graphs with moderate degrees. We show that in time O(Δ+log*n), a (Δ+1)-coloring can be computed, a task for which the best previous algorithm required time O(Δ*log(Δ) + log*n). The same result holds for the problem of computing a maximal independent set.