Discussiones Mathematicae Graph Theory 33(2) (2013) 461-465
doi: 10.7151/dmgt.1679

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A Tight Bound on the Set Chromatic Number

Jean-Sébastien Sereni

CNRS
LORIA, Université Diderot
Nancy, France

Zelealem B. Yilma

Department of Mathematics
Addis Ababa University
Addis Ababa, Ethiopia

Abstract

We provide a tight bound on the set chromatic number of a graph in terms of its chromatic number. Namely, for all graphs G, we show that χs(G) ≥ ⎡log2 χ(G) ⎤+ 1, where χs(G) and χ(G) are the set chromatic number and the chromatic number of G, respectively. This answers in the affirmative a conjecture of Gera, Okamoto, Rasmussen and Zhang.

Keywords: chromatic number, set coloring, set chromatic number, neighbor, distinguishing coloring

2010 Mathematics Subject Classification: 05C15.

References

[1]G. Chartrand, F. Okamoto, C.W. Rasmussen, and P. Zhang, The set chromatic number of a graph, Discuss. Math. Graph Theory 29 (2009) 545--561, doi: 10.7151/dmgt.1463.
[2]G. Chartrand, F. Okamoto, and P. Zhang, Neighbor-distinguishing vertex colorings of graphs, J. Combin. Math. Combin. Comput. 74 (2010) 223--251.
[3]R. Gera, F. Okamoto, C. Rasmussen, and P. Zhang, Set colorings in perfect graphs, Math. Bohem. 136 (2011) 61--68.

Received 28 February 2012
Accepted 5 June 2012