Abstract
The k-Coloring problem is to decide whether a graph can be colored with at most k colors such that no two adjacent vertices receive the same color. The List k -Coloring problem requires in addition that every vertex u must receive a color from some given set L(u) ⊆ {1,…,k}. Let P n denote the path on n vertices, and G + H and rH the disjoint union of two graphs G and H and r copies of H, respectively. For any two fixed integers k and r, we show that List k -Coloring can be solved in polynomial time for graphs with no induced rP 1 + P 5, hereby extending the result of Hoàng, Kamiński, Lozin, Sawada and Shu for graphs with no induced P 5. Our result is tight; we prove that for any graph H that is a supergraph of P 1 + P 5 with at least 5 edges, already List 5-Coloring is NP-complete for graphs with no induced H. We also show that List k -Coloring is fixed parameter tractable in k + r on graphs with no induced rP 1 + P 2, and that k-Coloring restricted to such graphs allows a polynomial kernel when parameterized by k. Finally, we show that List k -Coloring is fixed parameter tractable in k for graphs with no induced P 1 + P 3.
This work has been supported by ANR Blanc AGAPE (ANR-09-BLAN-0159-03) and EPSRC (EP/G043434/1).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bacsó, G., Tuza, Z.: Dominating cliques in P 5-free graphs. Periodica Mathematica Hungarica 21, 303–308 (1990)
Bondy, J.A., Murty, U.S.R.: Graph Theory. Springer Graduate Texts in Mathematics, vol. 244 (2008)
Broersma, H., Fomin, F.V., Golovach, P.A., Paulusma, D.: Three Complexity Results on Coloring P k -Free Graphs. In: Fiala, J., Kratochvíl, J., Miller, M. (eds.) IWOCA 2009. LNCS, vol. 5874, pp. 95–104. Springer, Heidelberg (2009)
Broersma, H.J., Golovach, P.A., Paulusma, D., Song, J.: Updating the complexity status of coloring graphs without a fixed induced linear forest (manuscript)
Broersma, H.J., Golovach, P.A., Paulusma, D., Song, J.: Determining the chromatic number of triangle-free 2P 3-free graphs in polynomial time (manuscript)
Bruce, D., Hoàng, C.T., Sawada, J.: A Certifying Algorithm for 3-Colorability of P 5-Free Graphs. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 594–604. Springer, Heidelberg (2009)
Dabrowski, K., Lozin, V., Raman, R., Ries, B.: Colouring Vertices of Triangle-Free Graphs. In: Thilikos, D.M. (ed.) WG 2010. LNCS, vol. 6410, pp. 184–195. Springer, Heidelberg (2010)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979)
Golovach, P.A., Paulusma, D., Song, J.: 4-Coloring H-free graphs when H is small (manuscript)
Grötschel, M., Lovász, L., Schrijver, A.: Polynomial algorithms for perfect graphs. Ann. Discrete Math., Topics on Perfect Graphs 21, 325–356 (1984)
Hoàng, C.T., Kamiński, M., Lozin, V., Sawada, J., Shu, X.: Deciding k-colorability of P 5-free graphs in polynomial time. Algorithmica 57, 74–81 (2010)
Holyer, I.: The NP-completeness of edge-coloring. SIAM J. Comput. 10, 718–720 (1981)
Jansen, K., Scheffler, P.: Generalized coloring for tree-like graphs. Discrete Appl. Math. 75, 135–155 (1997)
Kamiński, M., Lozin, V.V.: Coloring edges and vertices of graphs without short or long cycles. Contributions to Discrete Math. 2, 61–66 (2007)
Kamiński, M., Lozin, V.V.: Vertex 3-colorability of Claw-free Graphs. Algorithmic Operations Research 21 (2007)
Král’, D., Kratochvíl, J., Tuza, Z., Woeginger, G.J.: Complexity of coloring graphs without forbidden induced subgraphs. In: Brandstädt, A., Le, V.B. (eds.) WG 2001. LNCS, vol. 2204, pp. 254–262. Springer, Heidelberg (2001)
Kratochvíl, J.: Precoloring extension with fixed color bound. Acta Math. Univ. Comen. 62, 139–153 (1993)
Le, V.B., Randerath, B., Schiermeyer, I.: On the complexity of 4-coloring graphs without long induced paths. Theoret. Comput. Sci. 389, 330–335 (2007)
Leven, D., Galil, Z.: NP completeness of finding the chromatic index of regular graphs. Journal of Algorithms 4, 35–44 (1983)
Maffray, F., Preissmann, M.: On the NP-completeness of the k-colorability problem for triangle-free graphs. Discrete Math. 162, 313–317 (1996)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and its Applications. Oxford University Press (2006)
Randerath, B., Schiermeyer, I.: 3-Colorability ∈ P for P 6-free graphs. Discrete Appl. Math. 136, 299–313 (2004)
Randerath, B., Schiermeyer, I.: Vertex colouring and forbidden subgraphs - a survey. Graphs Combin. 20, 1–40 (2004)
Schindl, D.: Some new hereditary classes where graph coloring remains NP-hard. Discrete Math. 295, 197–202 (2005)
Tuza, Z.: Graph colorings with local restrictions - a survey. Discuss. Math. Graph Theory 17, 161–228 (1997)
Woeginger, G.J., Sgall, J.: The complexity of coloring graphs without long induced paths. Acta Cybernet. 15, 107–117 (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Couturier, JF., Golovach, P.A., Kratsch, D., Paulusma, D. (2011). List Coloring in the Absence of a Linear Forest. In: Kolman, P., Kratochvíl, J. (eds) Graph-Theoretic Concepts in Computer Science. WG 2011. Lecture Notes in Computer Science, vol 6986. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25870-1_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-25870-1_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-25869-5
Online ISBN: 978-3-642-25870-1
eBook Packages: Computer ScienceComputer Science (R0)