Advertisement

Journal of Combinatorial Optimization

, Volume 10, Issue 3, pp 211–225 | Cite as

On Split-Coloring Problems

  • T. Ekim
  • D. de Werra
Article

Abstract

We study a new coloring concept which generalizes the classical vertex coloring problem in a graph by extending the notion of stable sets to split graphs. First of all, we propose the packing problem of finding the split graph of maximum size where a split graph is a graph G = (V,E) in which the vertex set V can be partitioned into a clique K and a stable set S. No condition is imposed on the edges linking vertices in S to the vertices in K. This maximum split graph problem gives rise to an associated partitioning problem that we call the split-coloring problem. Given a graph, the objective is to cover all his vertices by a least number of split graphs. Definitions related to this new problem are introduced. We mention some polynomially solvable cases and describe open questions on this area.

Keywords

split-coloring vertex covering by split graphs partitioning packing 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. C. Benzaken, P. Hammer, and D. de Werra, “Split graphs of Dilworth number 2,” Discrete Mathematics, vol. 55, pp. 123–127, 1985.CrossRefMathSciNetGoogle Scholar
  2. C. Berge, Graphes, Bordas, Paris, 1983.Google Scholar
  3. A. Brandstädt, V. Le, and T. Szymczak, “The complexity of some problems related to graph 3-colorability,” Discrete Applied Mathematics, vol. 89, pp. 59–73, 1998.MathSciNetGoogle Scholar
  4. Z.A. Chernyak and A. Chernyak, “Split dimension of graphs,” Discrete Mathematics, vol. 89, pp. 1–6, 1991.CrossRefMathSciNetGoogle Scholar
  5. D. de Werra, M. Demange, J. Monnot, and V. Paschos, “A hypocoloring model for batch scheduling,” Discrete Applied Mathematics, vol. 146, pp. 3–26, 2005.MathSciNetGoogle Scholar
  6. G. Dirac, “On rigid circuit graphs,” Abh. Math. Sem. Univ. Hamburg, no. 25, pp. 71–76, 1961.Google Scholar
  7. S. Földes and P. Hammer, “On split graphs and some related questions,” in Problèmes Combinatoires et Théorie des Graphes, Orsay, France, Colloques Internationnaux C.N.R.S. 260, 1976, pp. 139–140.Google Scholar
  8. S. Földes and P. Hammer, “Split graphs,” Congressum Numerantium, vol. 19, pp. 311–315, 1977.Google Scholar
  9. D. Fulkerson and O. Gross, “Incidence matrixes and interval graphs,” Pacific Journal of Math., vol. 15, pp. 835–855, 1965.MathSciNetGoogle Scholar
  10. F. Gavril, “Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independant set of a chordal graph,” SIAM J. Comput., vol. 1, pp. 180–187, 1972.CrossRefMATHMathSciNetGoogle Scholar
  11. E. Gourdin, M. Labbé, and H. Yaman, “Telecommunication and location,” in Facility location, Springer, Berlin, 2002, pp. 275–305.Google Scholar
  12. P. Hammer and B. Simeone, “The splittance of a graph,” Combinatorica, vol. 1, pp. 275–284, 1981.MathSciNetGoogle Scholar
  13. P. Hell, S. Klein, L. Nogueira, and F. Protti, “Partitioning chordal graphs into independent sets and cliques,” Discrete Applied Mathematics, vol. 141, pp. 185–194, 2004.CrossRefMathSciNetGoogle Scholar
  14. N. Mahadev and U. Peled, Threshold Graphs and Related Topics, Ann. Disc. Mat., North-Holland, vol. 56, 1995.Google Scholar
  15. R. Tarjan, “Depth first search and linear graph algorithms,” SIAM J. Comput., vol. 1, pp. 146–160, 1972.CrossRefMATHMathSciNetGoogle Scholar
  16. R.E. Tarjan and M. Yannakakis, “Addendum: Simple linear time algorithms to test chordality of graphs, test acyclicity of hypergraphs and selectively reduce acyclic hypergraphs,” SIAM J. Comput., vol. 14, no. 1, pp. 254–255, 1985.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Institute of Mathematics—ROSEEcole Polytechnique Fédérale de LausanneLausanne-EcublensSwitzerland

Personalised recommendations