b-Coloring of the Mycielskian of Regular Graphs

  • S. Francis RajEmail author
  • M. Gokulnath
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11394)


The b-chromatic number b(G) of a graph G is the maximum k for which G has a proper vertex coloring using k colors such that each color class contains at least one vertex adjacent to a vertex of every other color class. In this paper, we have mainly investigated on the b-chromatic number of the Mycielskian of regular graphs. In particular, we have obtained the exact value of the b-chromatic number of the Mycielskian of some classes of graphs. This includes a few families of regular graphs, graphs with \(b(G)=2\) and split graphs. In addition, we have found bounds for the b-chromatic number of the Mycielskian of some more families of regular graphs in terms of the b-chromatic number of their original graphs.


b-coloring b-chromatic number Mycielskian of graphs Regular graphs 

2000 AMS Subject Classification




For the first author, this research was supported by SERB DST Project, Government of India, File no: EMR/2016/007339. For the second author, this research was supported by UGC - BSR, Research Fellowship, Government of India.


  1. 1.
    Balakrishnan, R., Kavaskar, T.: b-coloring of Kneser graphs. Discret. Appl. Math. 160(1–2), 9–14 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Balakrishnan, R., Raj, S.F.: Bounds for the b-chromatic number of the Mycielskian of some families of graphs. Ars Comb. 122, 89–96 (2015)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Balakrishnan, R., Raj, S.F., Kavaskar, T.: b-coloring of Cartesian product of odd graphs. Ars Comb. 131, 285–298 (2017)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Francis, P., Raj, S.F.: On b-coloring of powers of hypercubes. Discret. Appl. Math. 225, 74–86 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hall, P.: On representatives of subsets. J. London Math. Soc. 1(1), 26–30 (1935)CrossRefGoogle Scholar
  6. 6.
    Irving, R.W., Manlove, D.F.: The b-chromatic number of a graph. Discret. Appl. Math. 91(1–3), 127–141 (1999)CrossRefGoogle Scholar
  7. 7.
    Kouider, M.: b-chromatic number of a graph, subgraphs and degrees. Rapport interne LRI 1392 (2004)Google Scholar
  8. 8.
    Kouider, M., El Sahili, A.: About b-colouring of regular graphs. Rapport de Recherche 1432 (2006)Google Scholar
  9. 9.
    Kratochvíl, J., Tuza, Z., Voigt, M.: On the b-chromatic number of graphs. In: Goos, G., Hartmanis, J., van Leeuwen, J., Kučera, L. (eds.) WG 2002. LNCS, vol. 2573, pp. 310–320. Springer, Heidelberg (2002). Scholar
  10. 10.
    Mycielski, J.: Sur le coloriage des graphs. Colloq. Math. 3(2), 161–162 (1955)MathSciNetCrossRefGoogle Scholar
  11. 11.
    West, D.B.: Introduction to Graph Theory. Prentice-Hall of India Private Limited, Delhi (2005)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsPondicherry UniversityPuducherryIndia

Personalised recommendations