The local nature of Δ-coloring and its algorithmic applications

Abstract

Given a connected graphG=(V, E) with |V|=n and maximum degree Δ such thatG is neither a complete graph nor an odd cycle, Brooks' theorem states thatG can be colored with Δ colors. We generalize this as follows: letG-v be Δ-colored; then,v can be colored by considering the vertices in anO(logΔ n) radius aroundv and by recoloring anO(logΔ n) length “augmenting path” inside it. Using this, we show that Δ-coloringG is reducible inO(log3 n/logΔ) time to (Δ+1)-vertex coloringG in a distributed model of computation. This leads to fast distributed algorithms and a linear-processorNC algorithm for Δ-coloring.

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A preliminary version of this paper appeared as part of the paper “Improved Distributed Algorithms for Coloring and Network Decomposition Problems”, in theProceedings of the ACM Symposium on Theory of Computing pages 581–592, 1992. This research was done when the authors were at the Computer Science Department of Cornell University. The research was supported in part by NSF PYI award CCR-89-96272 with matching funds from UPS and Sun Microsystems.

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Panconesi, A., Srinivasan, A. The local nature of Δ-coloring and its algorithmic applications. Combinatorica 15, 255–280 (1995). https://doi.org/10.1007/BF01200759

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Mathematics Subject Classification (1991)

  • 68 Q 22
  • 05 C 15
  • 68 R 10