Three complexity results on coloring Pk-free graphs

  • Authors:

  • Hajo Broersma;Fedor V. Fomin;Petr A. Golovach;DaniëL Paulusma

  • Affiliations:

  • School of Engineering and Computing Sciences, Durham University, DH1 3LE Durham, United Kingdom;Department of Informatics, University of Bergen, PB 7803, 5020 Bergen, Norway;Department of Informatics, University of Bergen, PB 7803, 5020 Bergen, Norway;School of Engineering and Computing Sciences, Durham University, DH1 3LE Durham, United Kingdom

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2013

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Abstract

We prove three complexity results on vertex coloring problems restricted to P"k-free graphs, i.e., graphs that do not contain a path on k vertices as an induced subgraph. First of all, we show that the pre-coloring extension version of 5-coloring remains NP-complete when restricted to P"6-free graphs. Recent results of Hoang et al. imply that this problem is polynomially solvable on P"5-free graphs. Secondly, we show that the pre-coloring extension version of 3-coloring is polynomially solvable for P"6-free graphs. This implies a simpler algorithm for checking the 3-colorability of P"6-free graphs than the algorithm given by Randerath and Schiermeyer. Finally, we prove that 6-coloring is NP-complete for P"7-free graphs. This problem was known to be polynomially solvable for P"5-free graphs and NP-complete for P"8-free graphs, so there remains one open case.