Locality based graph coloring

We study the problem of locality based graph coloring. This problem is motivated by the problem of assigning time slots for broadcast in mobile packet radio networks. This problem has also been studied in the context of distributed and parallel graph coloring [4, 6, 9, 8]. In this problem, one has to design a coloring algorithm that assigns a color to a vertex based on the label of the vertex and the labels on its neighbors. Linial proved an upper bound of O(A2 log n) and a lower bound of fl(log log n) on the number of colors needed to locally color an n-vertex graph with maximum vertex degree A [9, 8]. His main motivation was that repeated application of local coloring gives a fast algorithm for distributed coloring. He proved that one could get a A2 coloring in O(log* n) steps this way. In this paper we improve upon the bounds for the problem of local coloring. Using a new characterization in terms of a family of set systems we design a randomized algorithm for the problem and prove an upper bound of O(A. 2A log log n). An important question left open in Linial’s paper was the case of large A. The best lower bound was A + 1. Linial observed that a result of Erdos, Frankl and Furedi implied that his method cannot be applied to reduce the number of colors to below (A~2). We obtain lower bounds that match the upper bounds within a factor that is poly-logarithmic in terms of these bounds. Of particular interest we have very precise bounds for the case when A > 2+. These bounds are useful to obtain a heuristic estimate on the *Researchsupported in part by Ketan Mulmuley’s Packard Fellowship. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of tha Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. 25th ACM STOC ‘93-51931CA, LJSA