Abstract
We study the complexity of several coloring problems on graphs, parameterized by the treewidth t of the graph:
(1) The list chromatic number χ l (G) of a graph G is defined to be the smallest positive integer r, such that for every assignment to the vertices v of G, of a list L v of colors, where each list has length at least r, there is a choice of one color from each vertex list L v yielding a proper coloring of G. We show that the problem of determining whether χ l (G) ≤ r, the List Chromatic Number problem, is solvable in linear time for every fixed treewidth bound t. The method by which this is shown is new and of general applicability.
(2) The List Coloring problem takes as input a graph G, together with an assignment to each vertex v of a set of colors C v . The problem is to determine whether it is possible to choose a color for vertex v from the set of permitted colors C v , for each vertex, so that the obtained coloring of G is proper. We show that this problem is W[1]-hard, parameterized by the treewidth of G. The closely related Precoloring Extension problem is also shown to be W[1]-hard, parameterized by treewidth.
(3) An equitable coloring of a graph G is a proper coloring of the vertices where the numbers of vertices having any two distinct colors differs by at most one. We show that the problem is hard for W[1], parameterized by (t,r). We also show that a list-based variation, List Equitable Coloring is W[1]-hard for trees, parameterized by the number of colors on the lists.
This is a preview of subscription content, log in via an institution to check access.
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
Alon, N.: Restricted colorings of graphs. In: Walker, K. (ed.) Surveys in Combinatorics 1993. London Math. Soc. Lecture Notes Series, vol. 187, pp. 1–33. Cambridge Univ. Press, Cambridge (1993)
Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. J. Algorithms 12, 308–340 (1991)
Bodlaender, H.L., Fomin, F.V.: Equitable colorings of bounded treewidth graphs. Theoretical Computer Science 349, 22–30 (2005)
Bodlaender, H.L., Fellows, M., Langston, M., Ragan, M.A., Rosamond, F., Weyer, M.: Quadratic kernelization for convex recoloring of trees. In: TSDM 2000. LNCS, Springer, Heidelberg (2007)
Borie, R.B., Parker, R.G., Tovey, C.A.: Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively generated graph families. Algorithmica 7, 555–581 (1992)
Chor, B., Fellows, M., Ragan, M.A., Razgon, I., Rosamond, F., Snir, S.: Connected coloring completion for general graphs: algorithms and complexity. In: Proceedings COCOON 2007. LNCS, Springer, Heidelberg (to appear)
Courcelle, B.: The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation 85, 12–75 (1990)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)
Erdös, P., Rubin, A.L., Taylor, H.: Choosability in graphs. Congressus Numerantium 26, 122–157 (1980)
Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)
Fellows, M., Giannopoulos, P., Knauer, C., Paul, C., Rosamond, F., Whitesides, S., Yu, N.: The lawnmower and other problems: applications of MSO logic in geometry. Manuscript (2007)
Fellows, M., Hermelin, D., Rosamond, F.: On the fixed-parameter intractability and tractability of multiple-interval graph properties. Manuscript (2007)
Jansen, K., Scheffler, P.: Generalized colorings for tree-like graphs. Discrete Applied Mathematics 75, 135–155 (1997)
Jensen, T.R., Toft, B.: Graph Coloring Problems. Wiley Interscience, Chichester (1995)
Kostochka, A.V., Pelsmajer, M.J., West, D.B.: A list analogue of equitable coloring. Journal of Graph Theory 44, 166–177 (2003)
Kratochvil, J., Tuza, Z., Voigt, M.: New trends in the theory of graph colorings: choosability and list coloring. In: Graham, R., et al. (eds.) Contemporary Trends in Discrete Mathematics (from DIMACS and DIMATIA to the future), AMS, Providence. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 49, pp. 183–197 (1999)
Marx, D.: Graph coloring with local and global constraints. Ph.D. dissertation, Department of Computer Science and Information Theory, Budapest University of Technology and Economics (2004)
Meyer, W.: Equitable coloring. American Mathematical Monthly 80, 920–922 (1973)
Niedermeier, R.: Invitation to Fixed Parameter Algorithms. Oxford University Press, Oxford (2006)
Thomassen, C.: Every planar graph is 5-choosable. J. Combinatorial Theory Ser. B 62, 180–181 (1994)
Tuza, Z.: Graph colorings with local constraints — A survey. Discussiones Mathematicae – Graph Theory 17, 161–228 (1997)
Vizing, V.G.: Coloring the vertices of a graph in prescribed colors (in Russian). Metody Diskret. Anal. v Teorii Kodov i Schem 29, 3–10 (1976)
Woodall, D.R.: List colourings of graphs. In: Hirschfeld, J.W.P. (ed.) Surveys in Combinatorics 2001. London Math. Soc. Lecture Notes Series, vol. 288, pp. 269–301. Cambridge Univ. Press, Cambridge (2001)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fellows, M. et al. (2007). On the Complexity of Some Colorful Problems Parameterized by Treewidth. In: Dress, A., Xu, Y., Zhu, B. (eds) Combinatorial Optimization and Applications. COCOA 2007. Lecture Notes in Computer Science, vol 4616. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73556-4_38
Download citation
DOI: https://doi.org/10.1007/978-3-540-73556-4_38
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73555-7
Online ISBN: 978-3-540-73556-4
eBook Packages: Computer ScienceComputer Science (R0)