The local nature of Δ-coloring and its algorithmic applications
Received: 06 October 1992 DOI:
10.1007/BF01200759 Cite this article as: Panconesi, A. & Srinivasan, A. Combinatorica (1995) 15: 255. doi:10.1007/BF01200759 Abstract
Given a connected graph
G=( V, E) with | V|= n and maximum degree Δ such that G is neither a complete graph nor an odd cycle, Brooks' theorem states that G can be colored with Δ colors. We generalize this as follows: let G- v be Δ-colored; then, v can be colored by considering the vertices in an O(log Δ n) radius around v and by recoloring an O(log Δ n) length “augmenting path” inside it. Using this, we show that Δ-coloring G is reducible in O(log 3 n/logΔ) time to (Δ+1)-vertex coloring G in a distributed model of computation. This leads to fast distributed algorithms and a linear-processor NC algorithm for Δ-coloring. Mathematics Subject Classification (1991) 68 Q 22 05 C 15 68 R 10
A preliminary version of this paper appeared as part of the paper “Improved Distributed Algorithms for Coloring and Network Decomposition Problems”, in the
Proceedings 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. References
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