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On the b-Coloring of Cographs and P 4-Sparse Graphs

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Abstract

A b-coloring of a graph is a coloring such that every color class admits a vertex adjacent to at least one vertex receiving each of the colors not assigned to it. The b-chromatic number of a graph G, denoted by χ b (G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is b-continuous if it admits a b-coloring with t colors, for every \(t = \chi(G), \ldots, \chi_b(G)\) . We define a graph G to be b-monotonic if χ b (H 1) ≥ χ b (H 2) for every induced subgraph H 1 of G, and every induced subgraph H 2 of H 1. In this work, we prove that P 4-sparse graphs (and, in particular, cographs) are b-continuous and b-monotonic. Besides, we describe a dynamic programming algorithm to compute the b-chromatic number in polynomial time within these graph classes.

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References

  1. Blidia, M., Ikhlef-Eschouf, N., Maffray, F.: Caractérisation des graphes bγ-parfaits. In: Actes du 4eme Colloque sur l’Optimisation et les Systèmes d’Information (COSI’2007), pp. 179–190, Oran, Algeria, 2007. An English version is given in [2] below

  2. Blidia, M., Ikhlef-Eschouf, N., Maffray, F.: Characterization of bγ-perfect graphs. Cahiers Leibniz 171, July 2008, http://www.g-scop.inpg.fr/CahiersLeibniz/2008/171/171.html

  3. Bodlaender, H.L.: Achromatic number is NP-complete for cographs and interval graphs. Inf. Proc. Lett., 31, 135–138 (1989)

    Google Scholar 

  4. Corneil, D., Lerchs, H., Stewart Burlingham, L.: Complement reducible graphs. Disc. Appl. Math., 3(3), 163–174 (1981)

  5. Corneil, D., Perl, Y., Stewart, L.: Cographs: recognition, applications and algorithms. Cong. Numer., 43, 249–258 (1984)

    Google Scholar 

  6. Corteel, S., Valencia-Pabon, M., Vera, J.: On approximating the b-chromatic number. Disc. Appl. Math., 146 (1), 618–622 (2005)

    Google Scholar 

  7. Effantin, B., Kheddouci, H.: The b-chromatic number of some power graphs. Disc. Math. & Theor. Comput. Sci., 6(1), 45–54 (2003)

    Google Scholar 

  8. Faik, T.: La b-continuité des b-colorations : complexité, propriétés structurelles et algorithmes. PhD thesis, L.R.I., Université Paris-Sud, Orsay, France (2005)

  9. Hoàng, C.T., Kouider, M.: On the b-dominating coloring of graphs. Disc. App. Math., 152, 176–186 (2005)

    Google Scholar 

  10. Hoàng, C.T., Linhares Sales, C., Maffray, F.: On minimally b-imperfect graphs. manuscript (2006)

  11. Hoàng, C.T.: Perfect graphs. PhD thesis, School of Computer Science, McGill University, Montreal (1985)

  12. Irving, R.W., Manlove, D.F.: The b-chromatic number of a graph. Disc. Appl. Math., 91, 127–141 (1999)

    Google Scholar 

  13. Jamison, B., Olariu, S.: Recognizing P 4-sparse graphs in linear time. SIAM J. on Comput., 21, 381–406 (1992)

  14. Jamison, B., Olariu, S.: A tree representation for P 4-sparse graphs. Disc. Appl. Math., 35, 115–129 (1992)

    Google Scholar 

  15. Jamison, B., Olariu, S.: Linear-time optimization algorithms for p 4-sparse graphs. Disc. Appl. Math., 61, 155–175 (1995)

    Google Scholar 

  16. Jansen, K., Scheffler, P., Woeginger, G.J.: Maximum covering with D cliques. Lect. Notes Comput. Sci., 710, 319–328 (1993)

    Google Scholar 

  17. Kára, J., Kratochvíl, J., Voigt, M.: b-continuity. Technical Report M 14/04, Technical University Ilmenau, Faculty of Mathematics and Natural Sciences (2004)

  18. Klein, S., Kouider, M.: On b-perfect graphs. In: Annals of the XII Latin-Ibero-American Congress on Operations Research, Havanna, Cuba, October 2004

  19. Kouider, M., Zaker, M.: Bounds for the b-chromatic number of some families of graphs. Dis. Math., 306, 617–623 (2006)

    Google Scholar 

  20. Kratochvíl, J., Tuza, Zs., Voigt, M.: On the b-chromatic number of a graph. Lect. Notes Comput. Sci., 2573, 310–320 (2002)

    Google Scholar 

  21. Maffray, F., Mechebbek, M.: On b-perfect chordal graphs. manuscript (2007)

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Correspondence to Flavia Bonomo.

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Flavia Bonomo: Partially supported by ANPCyT PICT-2007-00533 and PICT-2007-00518, and UBACyT Grants X069 and X606 (Argentina).

Guillermo Durán: Partially supported by FONDECyT Grant 1080286 and Millennium Science Institute “Complex Engineering Systems” (Chile), and ANPCyT PICT-2007-00518 and UBACyT Grant X069 (Argentina).

Javier Marenco: Partially supported by ANPCyT PICT-2007-00518 and UBACyT Grant X069 (Argentina).

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Bonomo, F., Durán, G., Maffray, F. et al. On the b-Coloring of Cographs and P 4-Sparse Graphs. Graphs and Combinatorics 25, 153–167 (2009). https://doi.org/10.1007/s00373-008-0829-1

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  • DOI: https://doi.org/10.1007/s00373-008-0829-1

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