Generalized chromatic numbers of random graphs

Let p be a hereditary graph property. The p ‐chromatic number of a graph is the minimal number of classes in a vertex partition wherein each class spans a subgraph with property p. For the property p of edgeless graphs the p ‐chromatic number is just the usual chromatic number, whose value is known to be (1/2 + o(1)) n /log 2 n for almost every graph of order n. We show that we may associate with every nontrivial hereditary property p an explicitly defined natural number r = r ( p ), and that the p ‐chromatic number is then (l/2r + o (1)) n /log 2 n for almost every graph of order n. © 1995 John Wiley & Sons, Inc.