Narrowing down the gap on the complexity of coloring Pk-free graphs

Authors:
Hajo Broersma;Petr A. Golovach;Daniël Paulusma;Jian Song
Affiliations:
School of Engineering and Computing Sciences, Durham University, Science Laboratories, Durham, UK;School of Engineering and Computing Sciences, Durham University, Science Laboratories, Durham, UK;School of Engineering and Computing Sciences, Durham University, Science Laboratories, Durham, UK;School of Engineering and Computing Sciences, Durham University, Science Laboratories, Durham, UK
Venue:
WG'10 Proceedings of the 36th international conference on Graph-theoretic concepts in computer science
Year:
2010

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Abstract

A graph is Pk-free if it does not contain an induced subgraph isomorphic to a path on k vertices. We show that deciding whether a P8- free graph can be colored with at most four colors is an NP-complete problem. This improves a result of Le, Randerath, and Schiermeyer, who showed that 4-coloring is NP-complete for P9-free graphs, and a result of Woeginger and Sgall, who showed that 5-coloring is NP-complete for P8-free graphs. Additionally, we prove that the pre-coloring extension version of 4-coloring is NP-complete for P7-free graphs, but that the pre-coloring extension version of 3-coloring is polynomially solvable for (P2 + P4)-free graphs, a subclass of P7-free graphs.