Discussiones Mathematicae Graph Theory 31(4) (2011) 709-735
doi: 10.7151/dmgt.1575

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

Mais Alkhateeb

Faculty of Mathematics and Computer Science
TU Bergakademie Freiberg
09596 Freiberg, Germany

Anja Kohl

Faculty of Mathematics and Computer Science
TU Bergakademie Freiberg
09596 Freiberg, Germany

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.

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Received 15 April 2010
Revised 9 November 2010
Accepted 11 November 2010