View More View Less
  • 1 Pondicherry University, Pondicherry–605014, India
  • 2 CIT campus, Taramani, Chennai-600113, India
Restricted access

Purchase article

USD  $25.00

1 year subscription (Individual Only)

USD  $800.00

A b-coloring of a graph G with k colors is a proper coloring of G using k colors in which each color class contains a color dominating vertex, that is, a vertex which has a neighbor in each of the other color classes. The largest positive integer k for which G has a b-coloring using k colors is the b-chromatic number b(G) of G. It was conjectured in [10], that for any two graphs G and H, b(G[H]) ≦ b(G) − 1|V (H)| + Δ(H) + 1 and b(GH) ≦ max {b(G)(Δ(H) + 1), b(H) Δ(G) + 1)}, where G[H] and GH denotes the lexicographic product and the strong product of G and H, respectively. In this paper, we disprove both conjectures.

  • [1]

    Balakrishnan, R. and Francis Raj, S., Bounds for the b-chromatic number of G - v, Discrete Appl. Math., 161(9) (2013), 11731179.

  • [2]

    Balakrishnan, R., Francis Raj, S. and Kavaskar, T., Bounds for the b-chromatic number of Cartesian product of graphs, Graphs Combin., 30 (2014), 511520.

    • Search Google Scholar
    • Export Citation
  • [3]

    Balakrishnan, R., Francis Raj, S. and Kavaskar, T., b-coloring of cartesian product of odd graphs, to appear in ARS Combinatoria.

  • [4]

    Balakrishnan, R. and Kavaskar, T., b-coloring of Kneser graphs, Discrete Appl. Math., 160 (2012), 914.

  • [5]

    Bonomo, F., Duran, G., Maffray, F., Marenco, J. and Valencia-Pabon, M., On the b-coloring of Cographs and P 4-Sparse Graphs, Graphs Combin., 25 (2009), 153167.

    • Search Google Scholar
    • Export Citation
  • [6]

    Effantin, B. and Kheddouci, H., The b-chromatic number of some power graphs, Discrete Math. Theor. Comput. Sci., 6 (2003), 4554.

  • [7]

    Harary, F. and Hedetniemi, S., The achromatic number of a graph, J. Combin. Theory., 8 (1970), 154161.

  • [8]

    Irving, R. W. and Manlove, D. F., The b-chromatic number of a graph, Discrete Appl. Math., 91 (1999), 127141.

  • [9]

    Javadi, R. and Omoomi, B., On the b-coloring of Cartesian product of graphs, Ars Combin., 107 (2012), 521536.

  • [10]

    Jakovac, M. and Peterin, I., On the b-chromatic number of some graph products, Studia Sci. Math. Hungar., 49 (2012), 156169.

  • [11]

    Kratochvil, J., Tuza, Z. and Voigt, M., On the b-chromatic number of graphs, Lecture Notes in Comput. Sci. 2573 (2002), 310320.

  • [12]

    Kouider, M. and Mahéo, M., Some bounds for the b-chromatic number of a graph, Discrete Math., 256 (2002), 267277.

  • [13]

    Kouider, M. and Mahéo, M., The b-chromatic number of the Cartesian product of two graphs, Studia Sci. Math. Hungar., 44 (2007), 4955.

    • Search Google Scholar
    • Export Citation
  • [14]

    Kouider, M. and Zaker, M., Bounds for the b-chromatic number of some families of graphs, Discrete Math., 306 (2006), 617623.

The author instruction is available in PDF.

Please, download the file from HERE

Manuscript submission: HERE

 

  • Impact Factor (2019): 0.486
  • Scimago Journal Rank (2019): 0.234
  • SJR Hirsch-Index (2019): 23
  • SJR Quartile Score (2019): Q3 Mathematics (miscellaneous)
  • Impact Factor (2018): 0.309
  • Scimago Journal Rank (2018): 0.253
  • SJR Hirsch-Index (2018): 21
  • SJR Quartile Score (2018): Q3 Mathematics (miscellaneous)

Language: English, French, German

Founded in 1966
Publication: One volume of four issues annually
Publication Programme: 2020. Vol. 57.
Indexing and Abstracting Services:

  • CompuMath Citation Index
  • Mathematical Reviews
  • Referativnyi Zhurnal/li>
  • Research Alert
  • Science Citation Index Expanded (SciSearch)/li>
  • SCOPUS
  • The ISI Alerting Services

 

Subscribers can access the electronic version of every printed article.

Senior editors

Editor(s)-in-Chief: Pálfy Péter Pál

Managing Editor(s): Sági, Gábor

Editorial Board

  • Biró, András (Number theory)
  • Csáki, Endre (Probability theory and stochastic processes, Statistics)
  • Domokos, Mátyás (Algebra (Ring theory, Invariant theory))
  • Győri, Ervin (Graph and hypergraph theory, Extremal combinatorics, Designs and configurations)
  • O. H. Katona, Gyula (Combinatorics)
  • Márki, László (Algebra (Semigroup theory, Category theory, Ring theory))
  • Némethi, András (Algebraic geometry, Analytic spaces, Analysis on manifolds)
  • Pach, János (Combinatorics, Discrete and computational geometry)
  • Rásonyi, Miklós (Probability theory and stochastic processes, Financial mathematics)
  • Révész, Szilárd Gy. (Analysis (Approximation theory, Potential theory, Harmonic analysis, Functional analysis))
  • Ruzsa, Imre Z. (Number theory)
  • Soukup, Lajos (General topology, Set theory, Model theory, Algebraic logic, Measure and integration)
  • Stipsicz, András (Low dimensional topology and knot theory, Manifolds and cell complexes, Differential topology)
  • Szász, Domokos (Dynamical systems and ergodic theory, Mechanics of particles and systems)
  • Tóth, Géza (Combinatorial geometry)

STUDIA SCIENTIARUM MATHEMATICARUM HUNGARICA
Gábor Sági
Address: P.O. Box 127, H–1364 Budapest, Hungary
Phone: (36 1) 483 8344 ---- Fax: (36 1) 483 8333
E-mail: smh.studia@renyi.mta.hu