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We develop the first parallel graph coloring heuristics with strongtheoretical guarantees on work and depth and coloring quality. The key idea isto design a relaxation of the vertex degeneracy order, a well-known graphtheory concept, and to color vertices in the order dictated by this relaxation.This introduces a tunable amount of parallelism into the degeneracy orderingthat is otherwise hard to parallelize. This simple idea enables significantbenefits in several key aspects of graph coloring. For example, one of ouralgorithms ensures polylogarithmic depth and a bound on the number of usedcolors that is superior to all other parallelizable schemes, while maintainingwork-efficiency. In addition to provable guarantees, the developed algorithmshave competitive run-times for several real-world graphs, while almost alwaysproviding superior coloring quality. Our degeneracy ordering relaxation is ofseparate interest for algorithms outside the context of coloring.

High-Performance Parallel Graph Coloring with Strong Guarantees on Work, Depth, and Quality