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Graphs and Combinatorics

, Volume 35, Issue 6, pp 1633–1646 | Cite as

On Graph Fall-Coloring: Existence and Constructions

  • Hemanshu KaulEmail author
  • Christodoulos Mitillos
Original Paper

Abstract

Graph fall-coloring, also known as idomatic partition or independent domatic partition of graphs, was formally introduced by Dunbar, Hedetniemi, Hedetniemi, Jacobs, Knisely, Laskar and Rall in 2000 as an extension of both Graph Coloring and Graph Domination. It asks for a partition of the vertex set of a given graph into independent dominating sets, or equivalently into maximal independent sets. We study two fundamental questions related to this concept: when such a partition of vertices can exist, and how it relates to a proper coloring. We construct graphs with a large number of possible fall-colorings and as a consequence of that we answer a question of Dunbar et al. (J Combin Math Combin Comput 33:257–273, 2000) by constructing a family of graphs with arbitrarily far apart chromatic number and fall-chromatic number. In fact, we construct graphs whose Fall set (collection of k such that the graph has a fall coloring with k colors) is an arbitrarily long arithmetic sequence, thus giving us graphs with Fall set of large order having large gaps between its elements. We give a sufficient condition on the minimum degree for fall-colorable graphs and characterize the sharpness of this bound. Related to this, we also construct families of graphs which have both a k-fall-coloring and a k-coloring that is not a fall coloring for every k, illustrating the complex relationship between idomatic and non-domatic independent partitions.

Keywords

Fall-coloring Idomatic partition Independent domatic partition Chromatic number Graph products 

Notes

References

  1. 1.
    Allan, R.B., Laskar, R.: On domination and independent domination numbers of a graph. Discr. Math. 23, 73–76 (1978)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Berge, C.: Theory of Graphs and its Applications. Methuen, London (1962)zbMATHGoogle Scholar
  3. 3.
    Bollobás, B.: Extremal Graph Theory. LMS Monographs, vol. 11. Academic Press, London (1978)zbMATHGoogle Scholar
  4. 4.
    Cockayne, E.J., Hedetniemi, S.T.: Disjoint independent dominating sets in graphs. Discr. Math. 15, 213–222 (1976)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cockayne, E.J., Hedetniemi, S.T., Miller, D.J.: Properties of hereditary hypergraphs and middle graphs. Can. Math. Bull. 21, 461–468 (1978)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Crescenzi, P., Kann, V., Karpinski, M.: A compendium of NP optimization problems. http://www.nada.kth.se/~viggo/wwwcompendium/. Accessed 27 Mar 2019
  7. 7.
    Dunbar, J.E., Hedetniemi, S.M., Hedetniemi, S.T., Jacobs, D.P., Knisely, J., Laskar, R.C., Rall, D.F.: Fall colorings of graphs. J. Combin. Math. Combin. Comput. 33, 257–273 (2000)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Erdős, P., Hobbs, A.M., Payan, C.: Disjoint cliques and disjoint maximal independent sets of vertices in graphs. Discr. Math. 42(1), 57–61 (1982)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Goddard, W., Henning, M.A.: Independent domination in graphs: a survey and recent results. Discr. Math. 313(7), 839–854 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Greenwell, D., Lovász, L.: Applications of product coloring. Acta Math. Acad. Sci. Hungar. 25, 335–340 (1974)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs, pp. 67–106. Marcel Dekker, New York (1998)zbMATHGoogle Scholar
  12. 12.
    Harutyunyan, A., Horn, P., Verstraete, J.: Independent Dominating Sets in Graphs of Girth Five, Combinatorics, Probability and Computing, to appearGoogle Scholar
  13. 13.
    Henning, M.A., Löwenstein, C., Rautenbach, D.: Remarks about disjoint dominating sets. Discr. Math. 309(23–24), 6451–6458 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kaul, H., Mitillos, C.: On graph fall-coloring—operators and heredity. J. Combin. Math. Combin. Comput. 106, 125–151 (2018)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W., Bohlinger, J.D. (eds.) Complexity of Computer Computations. The IBM Research Symposia Series. Springer, Boston, MA (1972)Google Scholar
  16. 16.
    Klavžar, S., Mekiš, G.: On idomatic partitions of direct products of complete graphs. Graphs Combin. 27(5), 713–726 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Lyle, J., Drake, N., Laskar, R.: Independent domatic partitioning or fall-coloring of strongly chordal graphs. Congr. Numer. 172, 149–159 (2005)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Laskar, R., Lyle, J.: Fall coloring of bipartite graphs and cartesian products of graphs. Discr. Appl. Math. 157(2), 330–338 (2009)CrossRefGoogle Scholar
  19. 19.
    Valencia-Pabon, M.: Idomatic partitions of direct products of complete graphs. Discr. Math. 310(5), 1118–1122 (2010)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Valencia-Pabon, M.: On some problems on idomatic partitions and b-colorings of direct products of complete graphs. Math. Contemp. 39, 93–100 (2010)MathSciNetzbMATHGoogle Scholar
  21. 21.
    West, D.B.: Introduction to Graph Theory. Prentice Hall Inc., Upper Saddle River (1996)zbMATHGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Applied MathematicsIllinois Institute of TechnologyChicagoUSA

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