3-Colorability ∈ P for P6-free graphs

  • Authors:

  • Bert Randerath;Ingo Schiermeyer

  • Affiliations:

  • Institut für Informatik, Universität zu Köln, D-50969 Köln, Germany;Institut für Diskrete Mathematik und Algebra, TU Bergakademie Freiberg, D-09596 Freiberg, Germany

  • Venue:
  • Discrete Applied Mathematics - The 1st cologne-twente workshop on graphs and combinatorial optimization (CTW 2001)
  • Year:
  • 2004

Quantified Score

Hi-index 0.00

Visualization

Abstract

In this paper, we study a chromatic aspect for the class of P6-free graphs. Here, the focus of our interest are graph classes (defined in terms of forbidden induced subgraphs) for which the question of 3-colorability can be decided in polynomial time and, if so, a proper 3-coloring can be determined also in polynomial time. Note that the 3-colorability decision problem is a well-known NP-complete problem, even for special graph classes, e.g. for triangle- and K1,5-free graphs (Discrete Math. 162 (1-3) (1996) 313-317). Therefore, it is unlikely that there exists a polynomial algorithm deciding whether there exists a 3-coloring of a given graph in general. Our approach is based on an encoding of the problem with Boolean formulas making use of the existence of bounded dominating subgraphs. Together with a structural analysis of the nonperfect K4-free members of the graph class in consideration we obtain our main result that 3-colorability can be decided in polynomial time for the class of P6-free graphs.