Abstract
The distribution of the chromatic number on random graphsG n, p is quite sharply concentrated. For fixedp it concentrates almost surely in √n ω(n) consecutive integers where ω(n) approaches infinity arbitrarily slowly. If the average degreepn is less thann 1/6, it concentrates almost surely in five consecutive integers. Large deviation estimates for martingales are used in the proof.
Similar content being viewed by others
References
B. Bollobás andP. Erdős, Cliques in random graphs,Math. Proc. Cambridge Phil. Soc.,80 (1976). 419–427.
W. F. de la Vega, On the Chromatic number of sparge random graphs, n:Graph theory and combinatorics (ed. B. Bollobás) Academic Press, London, 1984, 321–328.
P. Erdős andJ. Spencer,Probabilistic Methods in Combinatorics, Academic Press, New York, 1974.
G. Grimmet andC. McDiarmid, On coloring random graphs,Math. Proc. Cambridge Phil. Soc.,77 (1985), 313–324.
S. Karlin andH. Taylor,First Course in Stochastic Processes, 2nd ed., Academic Press, New York, 1975.
V. Milman andG. Schechtman,Asymptotic theory of normed linear spaces, Lecture Notes in Mathematics, Springer.
S. Shamir andR. Upfal, Sequential and distributed graph coloring algorithms with performance analyses in random graph spaces.J. of Algorithms,5 (1984), 488–501.