A new algorithm for on-line coloring bipartite graphs

We first show that for any bipartite graph H with at most five vertices there exists an on-line competitive algorithm for the class of H-free bipartite graphs. We then analyze the performance of an on-line algorithm for coloring bipartite graphs on various subfamilies. The algorithm yields new upper bounds for the on-line chromatic number of bipartite graphs. We prove that the algorithm is on-line competitive for P(7)-free bipartite graphs, i.e., that do not contain an induced path on seven vertices. The number of colors used by the on-line algorithm for P(6)-free and P(7)-free bipartite graphs is, respectively, bounded by roughly twice and roughly eight times the on-line chromatic number. In contrast, it is known that there exists no competitive on-line algorithm to color P(6)-free (or P(7)-free) bipartite graphs, i.e., for which the number of colors is bounded by any function depending only on the chromatic number.