Abstract
A graph is k-choosable if it admits a proper coloring of its vertices for every assignment of k (possibly different) allowed colors to choose from for each vertex. It is NP-hard to decide whether a given graph is k-choosable for k ≥ 3, and this problem is considered strictly harder than the k-coloring problem. Only few positive results are known on input graphs with a given structure. Here, we prove that the problem is fixed parameter tractable on P 5-free graphs when parameterized by k. This graph class contains the well known and widely studied class of cographs. Our result is surprising since the parameterized complexity of k-coloring is still open on P 5-free graphs. To give a complete picture, we show that the problem remains NP-hard on P 5-free graphs when k is a part of the input.
This work is supported by the Research Council of Norway.
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References
Alon, N.: Restricted colorings of graphs. In: Surveys in combinatorics. London Math. Soc. Lecture Note Ser., vol. 187, pp. 1–33. Cambridge Univ. Press, Cambridge (1993)
Bacsó, G., Tuza, Z.: Dominating cliques in \(P\sb 5\)-free graphs. Period. Math. Hungar. 21, 303–308 (1990)
Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph classes: a survey. SIAM Monographs on Discrete Mathematics and Applications. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1999)
Corneil, D.G., Perl, Y., Stewart, L.K.: A linear recognition algorithm for cographs. SIAM J. Comput. 14, 926–934 (1985)
Downey, R.G., Fellows, M.R.: Parameterized complexity. Monographs in Computer Science. Springer, Heidelberg (1999)
Erdős, P., Rubin, A.L., Taylor, H.: Choosability in graphs. In: Proceedings of the West Coast Conference on Combinatorics, Graph Theory and Computing, Humboldt State Univ., Arcata, Calif, 1980, Utilitas Math., pp. 125–157 (1979)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)
Garey, M.R., Johnson, D.S.: Computers and intractability. W. H. Freeman and Co., San Francisco (1979)
Gutner, S., Tarsi, M.: Some results on (a:b)-choosability, CoRR, abs/0802.1338 (2008)
Hoàng, C.T., Kamiński, M., Lozin, V.V., Sawada, J., Shu, X.: A note on k-colorability of p5-free graphs. In: Ochmański, E., Tyszkiewicz, J. (eds.) MFCS 2008. LNCS, vol. 5162, pp. 387–394. Springer, Heidelberg (2008)
Jensen, T.R., Toft, B.: Graph Coloring Problems. Wiley Interscience, Hoboken (1995)
Král, D., Kratochvíl, J., Tuza, Z., Woeginger, G.J.: Complexity of coloring graphs without forbidden induced subgraphs. In: Brandstädt, A., Van Le, B. (eds.) WG 2001. LNCS, vol. 2204, pp. 254–262. Springer, Heidelberg (2001)
Mahadev, N.V.R., Roberts, F.S., Santhanakrishnan, P.: 3-choosable complete bipartite graphs, Technical Report 49-91, Rutgers University, New Brunswick, NJ (1991)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)
Randerath, B., Schiermeyer, I.: 3-colorability ∈ P for P 6-free graphs. Discrete Appl. Math. 136, 299–313 (2004); The 1st Cologne-Twente Workshop on Graphs and Combinatorial Optimization (CTW 2001)
Thomassen, C.: Every planar graph is 5-choosable. J. Combin. Theory Ser. B 62, 180–181 (1994)
Tuza, Z.: Graph colorings with local constraints—a survey. Discuss. Math. Graph Theory 17, 161–228 (1997)
van ’t Hof, P., Paulusma, D.: A new characterization of p6-free graphs. In: Hu, X., Wang, J. (eds.) COCOON 2008. LNCS, vol. 5092, pp. 415–424. Springer, Heidelberg (2008)
Vizing, V.G.: Coloring the vertices of a graph in prescribed colors. Diskret. Analiz 101, 3–10 (1976)
Woeginger, G.J., Sgall, J.: The complexity of coloring graphs without long induced paths. Acta Cybernet. 15, 107–117 (2001)
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Golovach, P.A., Heggernes, P. (2009). Choosability of P 5-Free Graphs. In: Královič, R., Niwiński, D. (eds) Mathematical Foundations of Computer Science 2009. MFCS 2009. Lecture Notes in Computer Science, vol 5734. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03816-7_33
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