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 Hoàng 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.
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Broersma, H., Fomin, F.V., Golovach, P.A., Paulusma, D. (2009). Three Complexity Results on Coloring P k -Free Graphs. In: Fiala, J., Kratochvíl, J., Miller, M. (eds) Combinatorial Algorithms. IWOCA 2009. Lecture Notes in Computer Science, vol 5874. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10217-2_12
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DOI: https://doi.org/10.1007/978-3-642-10217-2_12
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