# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2018-2019): c. 84%

# Discussiones Mathematicae Graph Theory

## 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.

## References

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