## Abstract

Kearns et al. introduced the Graph Coloring Problem to model dynamic conflict resolution in social networks. Players, represented by the nodes of a graph, consecutively update their color from a fixed set of colors with the prospect of finally choosing a color that differs from all neighbors choices. The players only react on local information (the colors of their neighbors) and do not communicate. The reader might think of radio stations searching for transmission frequencies which are not subject to interference from other stations. While Kearns et al. (see [10]) empirically examined how human players deal with such a situation, Chaudury et al. performed a theoretical study and showed that, under a simple, greedy and selfish strategy, the players find a proper coloring of the graph within time \(O\left (\log \left (\frac{n} {\delta } \right )\right )\) with probability ≥ 1 − *δ*, where *n* is the number of nodes in the network and *δ* is arbitrarily small. In other words, the graph is properly colored within *τ* steps and \(\tau < c\log \left (\frac{n} {\delta } \right )\) with high probability for some constant *c*. Previous estimates on the constant *c* are very large. In this chapter we substantially improve the analysis and upper time bound for the proper coloring, by combining ideas from search games and probability theory.

## References

- 1.S. Alpern, D.J. Reyniers: Spatial Dispersion as a Dynamic Coordination Problem,
*Theory and Decision*,**53**, p. 29–59, (2002).MathSciNetzbMATHCrossRefGoogle Scholar - 2.K. Chaudhuri, F. Chung Graham, M. Shoaib Jamall: A Network Coloring Game, WINE ’08, p. 522–530, (2008).Google Scholar
- 3.F. Chung Graham: Graph Theory in the Information Age,
*Notices of AMS*,**57**, no. 6, p. 726–732, (July 2010).Google Scholar - 4.B. Eisenberg: On the Expectation of the Maximum of IID Geometric Random Variables,
*Statistics & Probability Letters*,**78**, p. 135–143, (2008).MathSciNetzbMATHCrossRefGoogle Scholar - 5.B. Escoffier, L. Gourvés, J. Monnot: Strategic Coloring of a Graph,
*Algorithms and Complexity, Lecture Notes in Computer Science*, 6087, Springer, Berlin, p. 155–166 (2010)Google Scholar - 6.T.S. Ferguson:
*Game Theory*, part II.Google Scholar - 7.K. Hamza: The Smallest Uniform Upper Bound on the Distance Between the Mean and the Median of the Binomial and Poisson Distributions,
*Statistics & Probability Letters*,**23**, p. 21–25, (1995).MathSciNetzbMATHCrossRefGoogle Scholar - 8.W. Hoeffding: On the Distribution of the Number of Successes in Independent Trials,
*An. Math. Statistics*,**27**, p. 713–721, (1956).MathSciNetzbMATHCrossRefGoogle Scholar - 9.R. Kaas and J.M. Buhrman: Mean, Median and Mode in Binomial Distributions,
*Statistica Neerlandica***34**(1), p. 13–18, (1980).Google Scholar - 10.M. Kearns, S. Suri, N. Montfort: An Experimental Study of the Coloring Problem on Human Suject Networks,
*Science*,**313**(5788), p. 824–827, (2006)CrossRefGoogle Scholar - 11.T. L. Lai and H. Robbins: Maximally Dependent Random Variables,
*Proc.Nat.Acad.Sci.USA*,**73**, No. 2, p. 286–288, February 1976, Statistics.Google Scholar - 12.T. L. Lai and H. Robbins: A Class of Dependent Random Variables and their Maxima,
*Z. Wahrscheinlichkeitstheorie verw. Gebiete*,**42**, p. 89–111, (1978).MathSciNetzbMATHCrossRefGoogle Scholar - 13.M. Luby: Removing Randomness in Parallel Computation without a Processor Penalty,
*FOCS*, p. 162–173, (1988).Google Scholar - 14.P.N. Panagopoulou and P. G. Spirakis: A Game Theoretic Approach for Efficient Graph Coloring,
*ISAAC 2008*, LNCS 5369, p. 183–195, (2008).MathSciNetGoogle Scholar - 15.M. Shaked and J.G. Shanthikumar:
*Stochastic Orders and their Applications*. Springer, New York (2007).Google Scholar