Abstract
We consider a weighted version of the subcoloring problem that we call the hypocoloring problem: given a weighted graph G=(V,E;w) where w(v)≥ 0, the goal consists in finding a partition \({\cal S}=(S_1,\ldots,S_k)\) of the node set of G into hypostable sets and minimizing ∑\(_{i=1}^{k}\) w(S i ) where an hypostable S is a subset of nodes which generates a collection of node disjoint cliques K. The weight of S is defined as max { ∑ v ∈ K w(v)| K ∈ S}. Properties of hypocolorings are stated; complexity and approximability results are presented in some graph classes. The associated decision problem is shown to be NP-complete for bipartite graphs and triangle-free planar graphs with maximum degree 3. Polynomial algorithms are given for graphs with maximum degree 2 and for trees with maximum degree Δ.
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
Albertson, M.O., Jamison, R.E., Hedetniemi, S.T., Locke, S.C.: The subchromatic number of a graph. Discrete Math. 74, 33–49 (1989)
Beineke, L.W., White, A.T.: Selected topics in graph theory. Academic Press, London (1978)
Berge, C.: Graphs and Hypergraphs. North Holland, Amsterdam (1973)
Bodlaender, H.L., Jansen, K., Woeginger, G.J.: Scheduling with incompatible jobs. Discrete Appl. Math. 55, 219–232 (1994)
Borowiecki, M., Broere, I., Frick, M., Mihok, P., Semanisin, G.: Survey of hereditary properties of graphs. Discussiones Mathematicae-Graph Theory 17, 5–50 (1997)
Boudhar, M., Finke, G.: Scheduling on a batch machine with job compatibilities. Jorbel 40, 69–80 (2000)
Broersma, H., Fomin, F.V., Nešetřil, J., Woeginger, G.J.: More about subcolorings (extended abstract). In: Kučera, L. (ed.) WG 2002. LNCS, vol. 2573, pp. 68–79. Springer, Heidelberg (2002)
Brooks, R.L.: On colouring the nodes of a network. In: Proc. Cambridge Phil. Soc., vol. 37, pp. 194–197 (1941)
Brown, J.L., Corneil, D.G.: On generalized graph colorings. J. Graph Theory 11, 87–99 (1987)
Demange, M., de Werra, D., Monnot, J., Paschos, V.T.: Weighted node coloring: When stable sets are expensive. In: Kučera, L. (ed.) WG 2002. LNCS, vol. 2573, pp. 114–125. Springer, Heidelberg (2002)
Dillon, M.R.: Conditionnal coloring, Ph.D. thesis, University of Colorado at Denver (1998)
Feige, U., Kilian, J.: Zero knowledge and the chromatic number. J. Comput.System Sci. 57, 187–199 (1998)
Fiala, J., Jansen, K., Le, V.B., Seidel, E.: Graph subcolorings: Complexity and algorithms. In: Brandstädt, A., Le, V.B. (eds.) WG 2001. LNCS, vol. 2204, pp. 154–165. Springer, Heidelberg (2001)
Garey, M.R., Johnson, D.S.: Computers and intractability. a guide to the theory of NP-completeness. Freeman, CA (1979)
Grotzsch, H.: Ein dreifarbensatz fur dreikreisfreie netze auf der kugel, Wiss. Z. Martin Luther Univ. Halle-Wittenberg, Math. Naturwiss Reihe 8, 109–120 (1959)
Harary, F.: Conditional colorability in graphs, in Graphs and Applications. In: Harary, F., Maybee, J. (eds.) Proc. First Colo. Symp. graph theory. Wiley intersci., Publ, N.Y (1985)
Lovász, L.: On decomposition of graphs. Stud. Sci. Math. Hung. 1, 237–238 (1966)
Mutzel, P., Odenthal, T., Scharbrodt, M.: The thickness of graphs: A survey (1998)
Rendl, F.: On the complexity of decomposing matrices arising in satellite communication. Operations Research Letters 4, 5–8 (1985)
Schaefer, T.J.: The complexity of satisfiability problems. In: Proc. STOC, pp. 216–226 (1978)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
de Werra, D., Demange, M., Monnot, J., Paschos, V.T. (2004). The Hypocoloring Problem: Complexity and Approximability Results when the Chromatic Number Is Small. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2004. Lecture Notes in Computer Science, vol 3353. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30559-0_32
Download citation
DOI: https://doi.org/10.1007/978-3-540-30559-0_32
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-24132-4
Online ISBN: 978-3-540-30559-0
eBook Packages: Computer ScienceComputer Science (R0)