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Dominated Colorings of Graphs

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Abstract

In this paper, we introduce and study a new coloring problem of a graph called the dominated coloring. A dominated coloring of a graph \(G\) is a proper vertex coloring of \(G\) such that each color class is dominated by at least one vertex of \(G\). The minimum number of colors among all dominated colorings is called the dominated chromatic number, denoted by \(\chi _{dom}(G)\). In this paper, we establish the close relationship between the dominated chromatic number \(\chi _{dom}(G)\) and the total domination number \(\gamma _t(G)\); and the equivalence for triangle-free graphs. We study the complexity of the problem by proving its NP-completeness for arbitrary graphs having \(\chi _{dom}(G) \ge 4\) and by giving a polynomial time algorithm for recognizing graphs having \(\chi _{dom}(G) \le 3\). We also give some bounds for planar and star-free graphs and exact values for split graphs.

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References

  1. Appel, K., Haken, W.: Every planar map is four colorable. Bull. Am. Math. Soc. 82(5), 711–712 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arumugam, S., Raja Chandrasekar, K., Misra, N., Geevarghese, P., Saurabh, S.: Algorithmic aspects of dominator colorings in graphs. In: IWOCA’2011, LNCS 7056, pp. 19–30 (2011)

  3. Boumediene Merouane, H., Chellali, M.: On the dominator colorings in trees. Discuss. Math. Graph Theory 32(4), 677–683 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chellali, M., Maffray, F.: Dominator colorings in some classes of graphs. Graphs Comb. 28, 97–107 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chellali, M., Volkmann, L.: Relations between the lower domination parameters and the chromatic number of a graph. Discrete Math. 274(1–3), 1–8 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chen, G., Gyarfas, A., Schelp, R.: Vertex colorings with a distance restriction. Discrete Math. 191(1–3), 65–82 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gera, R.: On the dominator colorings in bipartite graphs. Inf. Technol. New Gen., 947–952 (2007)

  8. Gera, R.: On dominator colorings in graphs. Graph Theory Notes of New York LII, pp. 25–30 (2007)

  9. Gera, R., Horton, S., Rasmussen, C.: Dominator colorings and safe clique partitions. Congr. Numer. 181, 19–32 (2006)

    MATH  MathSciNet  Google Scholar 

  10. Haddad, M., Kheddouci, H.: A strict strong coloring of trees. Inf. Process. Lett. 109(18), 1047–1054 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Marcel Dekker, New York (1998)

    MATH  Google Scholar 

  12. Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in Graphs: Advanced Topics. Marcel Dekker, New York (1998)

    MATH  Google Scholar 

  13. Laskar, R., Pfaff, J., Hedetniemi, S.M., Hedetniemi, S.T.: On the algorithmic complexity of total domination. SIAM J. Alg. Discrete Methods 5(3), 420–425 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  14. Malaguti, E., Toth, P.: A survey on vertex coloring problems. Int. Trans. Oper. Res. 17, 1–34 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

We thank the referees for comments that helped us to improve the paper.

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Authors and Affiliations

  1. LAMDA-RO Laboratory, Department of Mathematics, University of Blida, B.P. 270, Blida, Algeria

    Houcine Boumediene Merouane & Mustapha Chellali

  2. Laboratoire LIRIS, UMR CNRS 5205, Université de Lyon, 69003 , Lyon, France

    Mohammed Haddad & Hamamache Kheddouci

  3. Université Claude Bernard Lyon 1, Campus de la doua, Bat. Nautibus, 43, Bd du 11 Novembre 1918, 69622 , Villeurbanne Cedex, France

    Mohammed Haddad & Hamamache Kheddouci

Authors
  1. Houcine Boumediene Merouane
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  2. Mohammed Haddad
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  3. Mustapha Chellali
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  4. Hamamache Kheddouci
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Corresponding author

Correspondence to Mohammed Haddad.

Additional information

This research was supported by “Region Rhône-Alpes COOPERA project” and “Programmes Nationaux de Recherche: Code 8/u09/510”.

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Merouane, H.B., Haddad, M., Chellali, M. et al. Dominated Colorings of Graphs. Graphs and Combinatorics 31, 713–727 (2015). https://doi.org/10.1007/s00373-014-1407-3

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  • DOI: https://doi.org/10.1007/s00373-014-1407-3

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