AbstractThe interval number of a graph G is the least natural number t such that G is the intersection graph of sets, each of which is the union of at most t intervals, denoted by i(G). Griggs and West showed that $i(G)\le \lceil {1\over 2} (d+1)\rceil $. We describe the extremal graphs for that inequality when d is even. For three special perfect graph classes we give bounds on the interval number in terms of the independence number. Finally, we show that a graph needs to contain large complete bipartite induced subgraphs in order to have interval number larger than the random graph on the same number of vertices. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 241–253, 2004