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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. Awerbuch, A. V. Goldberg, M. Luby, andS. A. Plotkin: Network decomposition locality in distributed computation, in:Proceedings of the IEEE Symposium on Foundations of Computer Science, (1989), 364–369.
B. Bollobás:Graph Theory, Springer Verlag, New York, 1979.
R. L. Brooks: On colouring the nodes of a network,Proc. Cambridge Phil. Soc. 37 (1941), 194–197.
R. Cole, andU. Vishkin: Deterministic coin tossing with applications to optimal parallel list ranking,Information and Control 70 (1986), 32–53.
A. V. Goldberg, S. A. Plotkin, andG. E. Shannon: Parallel symmetry-breaking in sparse graphs,SIAM J. Disc. Math. 1 (1989), 434–446.
P. Hajnal, andE. Szemerédi: Brooks coloring in parallel,SIAM. J. Disc. Math. 3 (1990), 74–80.
M. Krachmer, andJ. Naor: A faster parallel algorithm to color a graphy with Δ colors,Journal of Algorithms 9 (1988), 83–91.
H. J. Karloff: An NC-algorithm for Brooks' theorem,Theoretical Computer Science 68(1) (1989), 89–103.
R. M. Karp: Probabilistic recurrence relations, in:Proceedings of the ACM Symposium on Theory of Computing (1991), 190–197.
G. F. Lev, N. Pippenger, andL. G. Valiant: A fast parallel algorithm for routing in permutation networks,IEEE Transactions on Computers 30 (1981), 93–100.
N. Linial: Locality in distributed graph algorithms,SIAM J. Comput. 21(1) (1992), 193–201.
M. Luby: Removing randomness in parallel computation without a processor penalty,Journal of Computer and System Sciences,47 (1993), 250–286.
A. Panconesi, andA. Srinivasan: Improved distributed algorithm for coloring and network decomposition problems, in:Proceedings of the ACM Symposium on Theory of Computing (1992), 581–592.
Author information
Authors and Affiliations
Additional information
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.
Rights and permissions
About this article
Cite this article
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
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01200759