On the dominator colorings in trees
Houcine Boumediene Merouane ; Mustapha Chellali
Discussiones Mathematicae Graph Theory, Tome 32 (2012), p. 677-683 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

In a graph G, a vertex is said to dominate itself and all its neighbors. A dominating set of a graph G is a subset of vertices that dominates every vertex of G. The domination number γ(G) is the minimum cardinality of a dominating set of G. A proper coloring of a graph G is a function from the set of vertices of the graph to a set of colors such that any two adjacent vertices have different colors. A dominator coloring of a graph G is a proper coloring such that every vertex of V dominates all vertices of at least one color class (possibly its own class). The dominator chromatic number χd(G) is the minimum number of color classes in a dominator coloring of G. Gera showed that every nontrivial tree T satisfies γ(T)+1≤χd(T)≤γ(T)+2. In this note we characterize nontrivial trees T attaining each bound.

Publié le : 2012-01-01
EUDML-ID : urn:eudml:doc:270842 
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1635,
     author = {Houcine Boumediene Merouane and Mustapha Chellali},
     title = {On the dominator colorings in trees},
     journal = {Discussiones Mathematicae Graph Theory},
     volume = {32},
     year = {2012},
     pages = {677-683},
     zbl = {1293.05256},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1635}
}

Houcine Boumediene Merouane; Mustapha Chellali. On the dominator colorings in trees. Discussiones Mathematicae Graph Theory, Tome 32 (2012) pp. 677-683. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1635/

[000] [1] M. Chellali and F. Maffray, Dominator colorings in some classes of graphs, Graphs Combin. 28 (2012) 97-107, doi: 10.1007/s00373-010-1012-z. | Zbl 1234.05082

[001] [2] R. Gera, On the dominator colorings in bipartite graphs in: Proceedings of the 4th International Conference on Information Technology: New Generations (2007) 947-952, doi: 10.1109/ITNG.2007.142.

[002] [3] R. Gera, On dominator colorings in graphs, Graph Theory Notes of New York LII (2007) 25-30.

[003] [4] R. Gera, S. Horton and C. Rasmussen, Dominator colorings and safe clique partitions, Congr. Numer. 181 (2006) 19-32. | Zbl 1113.05032

[004] [5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, 1998). | Zbl 0890.05002

[005] [6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs: Advanced Topics (Marcel Dekker, Inc., New York, 1998). | Zbl 0883.00011

[006] [7] O. Ore, Theory of Graphs (Amer. Math. Soc. Colloq. Publ. 38, 1962).

[007] [8] L. Volkmann, On graphs with equal domination and covering numbers, Discrete Appl. Math. 51 (1994) 211-217, doi: 10.1016/0166-218X(94)90110-4. | Zbl 0803.05049