Coloring Random Graphs: A Short Survey

Abstract

We shortly present some recent results concerning the chromatic number of random graphs. The setting is as follows: we consider a probability space with graphs of a given average degree d and n vertices (the term "average" here refers to the ratio of the sum of the degrees of all vertices to n). In the first case (the Erd¿s-Renyi graphs), the probability space comprises all graphs with average degree d and n vertices. In the second case (regular graphs), the probability space comprises only graphs where all n vertices have degree exactly d. In both cases the probability measure is uniform. In both cases the chromatic number exhibits interesting threshold behavior: for a given average degree (given constant degree, for the regular case, respectively), the chromatic number of almost all graphs (asymptotically with n) lies within a common, small window of 1-3 integers. However as the degree increases, at specific values this window undergoes abrupt changes.