Friday, March 27, 2009 - 3:00pm
1 hour (actually 50 minutes)
Choose a graph uniformly at random from all d-regular graphs on n vertices. We determine the chromatic number of the graph for about half of all values of d, asymptotically almost surely (a.a.s.) as n grows large. Letting k be the smallest integer satisfying d < 2(k-1)\log(k-1), we show that a random d-regular graph is k-colorable a.a.s. Together with previous results of Molloy and Reed, this establishes the chromatic number as a.a.s. k-1 or k. If furthermore d>(2k-3)\log(k-1) then the chromatic number is a.a.s. k. This is an improvement upon results recently announced by Achlioptas and Moore. The method used is "small subgraph conditioning'' of Robinson and Wormald, applied to count colorings where all color classes have the same size. It is the first rigorously proved result about the chromatic number of random graphs that was proved by small subgraph conditioning. This is joint work with Xavier Perez-Gimenez and Nick Wormald.