Chapter

Algorithms and Complexity

Volume 3998 of the series Lecture Notes in Computer Science pp 284-295

On-Line Coloring of H-Free Bipartite Graphs

  • H. J. BroersmaAffiliated withDepartment of Computer Science, Durham University
  • , A. CapponiAffiliated withComputer Science, Division of Engineering and Applied Sciences, California Institute of Technology
  • , D. PaulusmaAffiliated withDepartment of Computer Science, Durham University

Abstract

We present a new on-line algorithm for coloring bipartite graphs. This yields a new upper bound on the on-line chromatic number of bipartite graphs, improving a bound due to Lovász, Saks and Trotter. The algorithm is on-line competitive on various classes of H – free bipartite graphs, in particular P 6-free bipartite graphs and P 7-free bipartite graphs, i.e., that do not contain an induced path on six, respectively seven vertices. The number of colors used by the on-line algorithm in these particular cases is bounded by roughly twice, respectively 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 only depending on the chromatic number.