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Three complexity results on coloring Pk-free graphs.

Broersma, H.J. and Fomin, F.V. and Golovach, P.A. and Paulusma, Daniel (2013) 'Three complexity results on coloring Pk-free graphs.', European journal of combinatorics., 34 (3). pp. 609-619.


We prove three complexity results on vertex coloring problems restricted to PkPk-free graphs, i.e., graphs that do not contain a path on kk 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 P6P6-free graphs. Recent results of Hoàng et al. imply that this problem is polynomially solvable on P5P5-free graphs. Secondly, we show that the pre-coloring extension version of 3-coloring is polynomially solvable for P6P6-free graphs. This implies a simpler algorithm for checking the 3-colorability of P6P6-free graphs than the algorithm given by Randerath and Schiermeyer. Finally, we prove that 6-coloring is NP-complete for P7P7-free graphs. This problem was known to be polynomially solvable for P5P5-free graphs and NP-complete for P8P8-free graphs, so there remains one open case.

Item Type:Article
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Publisher statement:NOTICE: this is the author’s version of a work that was accepted for publication in European journal of combinatorics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in European journal of combinatorics, 34, 3, 2013, 10.1016/j.ejc.2011.12.008
Date accepted:No date available
Date deposited:17 April 2013
Date of first online publication:April 2013
Date first made open access:No date available

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