PDF
Discussiones Mathematicae Graph Theory 31(4) (2011)
709-735
DOI: https://doi.org/10.7151/dmgt.1575
Upper bounds on the b-chromatic number and results for restricted graph classes
Mais Alkhateeb
Faculty of Mathematics and Computer Science | Anja Kohl
Faculty of Mathematics and Computer Science |
Abstract
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 χb(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 ≤ χb(G). In this paper, we establish four general upper bounds on χb(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 χb(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.Keywords: coloring, b-coloring, b-chromatic number, b-continuity
2010 Mathematics Subject Classification: 05C15, 05C78.
References
[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). |
Received 15 April 2010
Revised 9 November 2010
Accepted 11 November 2010
Close