On-Line and First-Fit Coloring of Graphs That Do Not Induce P5

For a graph H, let Forb(H) be the class of graphs that do not induce H, and let P5 be the path on five vertices. In this article, we answer two questions of Gyrfs and Lehel. First, we show that there exists a function f(w) such that for any graph G E Forb(Ph), the on-line coloring algorithm First-Fit uses at most f(w(G)) colors on G, where w(G) is the clique size of G. Second, we show that there exists an on-line algorithm A that will color any graph G E Forb(P5 with a number of colors exponential in w(G). Finally, we extend some of our results to larger classes of graphs defined in terms of a list of forbidden subgraphs.