PDF
Discussiones Mathematicae Graph Theory 33(2) (2013)
461-465
DOI: https://doi.org/10.7151/dmgt.1679
A Tight Bound on the Set Chromatic Number
Jean-Sébastien Sereni
CNRS | Zelealem B. Yilma
Department of Mathematics |
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
Close