, Volume 73, Issue 2, pp 289–305 | Cite as

\(b\)-Coloring is NP-hard on Co-bipartite Graphs and Polytime Solvable on Tree-Cographs

  • Flavia Bonomo
  • Oliver Schaudt
  • Maya Stein
  • Mario Valencia-Pabon


A b-coloring of a graph is a proper coloring such that every color class contains a vertex that is adjacent to all other color classes. The b-chromatic number of a graph \(G\), denoted by \(\chi _b(G)\), is the maximum number \(t\) such that \(G\) admits a b-coloring with \(t\) colors. A graph \(G\) is called b-continuous if it admits a b-coloring with \(t\) colors, for every \(t = \chi (G),\ldots ,\chi _b(G)\), and b-monotonic if \(\chi _b(H_1) \ge \chi _b(H_2)\) for every induced subgraph \(H_1\) of \(G\), and every induced subgraph \(H_2\) of \(H_1\). We investigate the b-chromatic number of graphs with stability number two. These are exactly the complements of triangle-free graphs, thus including all complements of bipartite graphs. The main results of this work are the following: (1) We characterize the b-colorings of a graph with stability number two in terms of matchings with no augmenting paths of length one or three. We derive that graphs with stability number two are b-continuous and b-monotonic. (2) We prove that it is NP-complete to decide whether the b-chromatic number of co-bipartite graph is at least a given threshold. (3) We describe a polynomial time dynamic programming algorithm to compute the b-chromatic number of co-trees. (4) Extending several previous results, we show that there is a polynomial time dynamic programming algorithm for computing the b-chromatic number of tree-cographs. Moreover, we show that tree-cographs are b-continuous and b-monotonic.


b-Coloring Stability number two Co-triangle-free graphs  NP-hardness Treecographs Polytime dynamic programming algorithms 



We would like to thank the anonymous referees for their careful reading and suggestions that helped us to improve the paper. This work was partially supported by UBACyT Grant 20020100100980, CONICET PIP 112-200901-00178 and 11220120100450CO, and ANPCyT PICT 2012–1324 (Argentina) and MathAmSud Project 13MATH-07 (Argentina–Brazil–Chile–France).


  1. 1.
    Berge, C.: Two theorems in graph theory. Proc. Natl. Acad. Sci. USA 43, 842–844 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bodlaender, H.L.: Achromatic number is NP-complete for cographs and interval graphs. Inf. Process. Lett. 31, 135–138 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bonomo, F., Durán, G., Maffray, F., Marenco, J., Valencia-Pabon, M.: On the b-coloring of cographs and \(P_4\)-sparse graphs. Graphs Comb. 25(2), 153–167 (2009)CrossRefzbMATHGoogle Scholar
  4. 4.
    Dantzig, G.B.: Discrete-variable extremum problems. Oper. Res. 5, 266–277 (1957)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Edmonds, J.: Paths, trees and flowers. Can. J. Math. 17, 449–467 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Faik, T.: La b-continuité des b-colorations: complexité, propriétés structurelles et algorithmes. Ph.D. thesis, L.R.I., Université Paris-Sud, Orsay, France (2005)Google Scholar
  7. 7.
    Harary, F., Hedetniemi, S.: The achromatic number of a graph. J. Comb. Theory 8, 154–161 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Havet, F., Linhares-Sales, C., Sampaio, L.: b-coloring of tight graphs. Discrete Appl. Math. 160(18), 2709–2715 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hoàng, C.T., Kouider, M.: On the b-dominating coloring of graphs. Discrete Appl. Math. 152, 176–186 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hoàng, C.T., Linhares Sales, C., Maffray, F.: On minimally b-imperfect graphs. Discrete Appl. Math. 157(17), 3519–3530 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Irving, R.W., Manlove, D.F.: The b-chromatic number of a graph. Discrete Appl. Math. 91, 127–141 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kára, J., Kratochvíl, J., Voigt, M.: b-Continuity. Technical Report M 14/04, Technical University Ilmenau, Faculty of Mathematics and Natural Sciences (2004)Google Scholar
  13. 13.
    Kratochvíl, J., Tuza, Zs, Voigt, M.: On the b-chromatic number of a graph. Lect. Notes Comput. Sci. 2573, 310–320 (2002)CrossRefGoogle Scholar
  14. 14.
    Tinhofer, G.: Strong tree-cographs are Birkoff graphs. Discrete Appl. Math. 22(3), 275–288 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Yannakakis, M., Gavril, F.: Edge dominating sets in graphs. SIAM J. Appl. Math. 38(3), 364–372 (1980)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Flavia Bonomo
    • 1
  • Oliver Schaudt
    • 2
  • Maya Stein
    • 3
  • Mario Valencia-Pabon
    • 4
    • 5
  1. 1.CONICET and Departamento de Computación, Facultad de Ciencias Exactas y NaturalesUniversidad de Buenos AiresBuenos AiresArgentina
  2. 2.Institut de Mathématiques de Jussieu, CNRS UMR7586Université Pierre et Marie Curie (Paris 6)ParisFrance
  3. 3.Centro de Modelamiento MatemáticoUniversidad de ChileSantiagoChile
  4. 4.LIPN, CNRS, UMR7030Université Paris 13, Sorbonne Paris CitéVilletaneuseFrance
  5. 5.INRIA Nancy - Grand EstNancyFrance

Personalised recommendations