Expose-and-merge exploration and the chromatic number of a random graph

Abstract

The expose-and-merge paradigm for exploring random graphs is presented. An algorithm of complexityn ^O(logn) is described and used to show that the chromatic number of a random graph for any edge probability 0<p<1 falls in the interval

$$\left[ {\left( {\frac{1}{2} - \varepsilon } \right)\log (1/(1 - p))\frac{n}{{\log n}}, \left( {\frac{2}{3} + \varepsilon } \right)\log (1/(1 - p))\frac{n}{{\log n}}} \right]$$

with probability approaching unity asn→∞.