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2011 | 31 | 4 | 709-735
Tytuł artykułu

Upper bounds on the b-chromatic number and results for restricted graph classes

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
A b-coloring of a graph G by k colors is a proper vertex coloring such that every color class contains a color-dominating vertex, that is, a vertex having neighbors in all other k-1 color classes. The b-chromatic number $χ_b(G)$ is the maximum integer k for which G has a b-coloring by k colors. Moreover, the graph G is called b-continuous if G admits a b-coloring by k colors for all k satisfying $χ(G) ≤ k ≤ χ_b(G)$. In this paper, we establish four general upper bounds on $χ_b(G)$. We present results on the b-chromatic number and the b-continuity problem for special graphs, in particular for disconnected graphs and graphs with independence number 2. Moreover we determine $χ_b(G)$ for graphs G with minimum degree δ(G) ≥ |V(G)|-3, graphs G with clique number ω(G) ≥ |V(G)|-3, and graphs G with independence number α(G) ≥ |V(G)|-2. We also prove that these graphs are b-continuous.
Słowa kluczowe
Wydawca
Rocznik
Tom
31
Numer
4
Strony
709-735
Opis fizyczny
Daty
wydano
2011
otrzymano
2010-04-15
poprawiono
2010-11-09
zaakceptowano
2010-11-11
Twórcy
  • Faculty of Mathematics and Computer Science, TU Bergakademie Freiberg, 09596 Freiberg, Germany
autor
  • Faculty of Mathematics and Computer Science, TU Bergakademie Freiberg, 09596 Freiberg, Germany
Bibliografia
  • [1] D. Barth, J. Cohen and T. Faik, On the b-continuity property of graphs, Discrete Appl. Math. 155 (2007) 1761-1768, doi: 10.1016/j.dam.2007.04.011.
  • [2] T. Faik and J.-F. Sacle, Some b-continuous classes of graphs, Technical Report N1350, LRI (Universite de Paris Sud, 2003).
  • [3] J.L. Gross and J. Yellen, Handbook of Graph Theory (CRC Press, 2004).
  • [4] C.T. Hoang and M. Kouider, On the b-dominating coloring of graphs, Discrete Appl. Math. 152 (2005) 176-186, doi: 10.1016/j.dam.2005.04.001.
  • [5] R.W. Irving and D.F. Manlove, The b-chromatic number of a graph, Discrete Appl. Math. 91 (1999) 127-141, doi: 10.1016/S0166-218X(98)00146-2.
  • [6] J. Kará, J. Kratochvil and M. Voigt, b-continuity, Preprint No. M 14/04, Technical University Ilmenau, Faculty for Mathematics and Natural Sciences (2004).
  • [7] A. Kohl and I. Schiermeyer, Some Results on Reed's Conjecture about ω, Δ, and χ with respect to α, Discrete Math. 310 (2010) 1429-1438, doi: 10.1016/j.disc.2009.05.025.
  • [8] M. Kouider and M. Maheo, Some bounds for the b-chromatic number of a graph, Discrete Math. 256 (2002) 267-277, doi: 10.1016/S0012-365X(01)00469-1.
  • [9] M. Kouider and M. Zaker, Bounds for the b-chromatic number of some families of graphs, Discrete Math. 306 (2006) 617-623, doi: 10.1016/j.disc.2006.01.012.
  • [10] L. Rabern, A note on Reed's conjecture, SIAM J. Discrete Math. 22 (2008) 820-827, doi: 10.1137/060659193.
  • [11] S. Radziszowski, Small Ramsey Numbers, Electronic Journal of Combinatorics, Dynamic Survey DS1 (2006).
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1575
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