# On the dominator colorings in trees

Houcine Boumediene Merouane; Mustapha Chellali

Discussiones Mathematicae Graph Theory (2012)

- Volume: 32, Issue: 4, page 677-683
- ISSN: 2083-5892

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topHoucine Boumediene Merouane, and Mustapha Chellali. "On the dominator colorings in trees." Discussiones Mathematicae Graph Theory 32.4 (2012): 677-683. <http://eudml.org/doc/270842>.

@article{HoucineBoumedieneMerouane2012,

abstract = {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 $χ_d(G)$ is the minimum number of color classes in a dominator coloring of G. Gera showed that every nontrivial tree T satisfies $γ(T)+1 ≤ χ_d(T) ≤ γ(T)+2$. In this note we characterize nontrivial trees T attaining each bound.},

author = {Houcine Boumediene Merouane, Mustapha Chellali},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {dominator coloring; domination; trees},

language = {eng},

number = {4},

pages = {677-683},

title = {On the dominator colorings in trees},

url = {http://eudml.org/doc/270842},

volume = {32},

year = {2012},

}

TY - JOUR

AU - Houcine Boumediene Merouane

AU - Mustapha Chellali

TI - On the dominator colorings in trees

JO - Discussiones Mathematicae Graph Theory

PY - 2012

VL - 32

IS - 4

SP - 677

EP - 683

AB - 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 $χ_d(G)$ is the minimum number of color classes in a dominator coloring of G. Gera showed that every nontrivial tree T satisfies $γ(T)+1 ≤ χ_d(T) ≤ γ(T)+2$. In this note we characterize nontrivial trees T attaining each bound.

LA - eng

KW - dominator coloring; domination; trees

UR - http://eudml.org/doc/270842

ER -

## References

top- [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. Zbl1234.05082
- [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.
- [3] R. Gera, On dominator colorings in graphs, Graph Theory Notes of New York LII (2007) 25-30.
- [4] R. Gera, S. Horton and C. Rasmussen, Dominator colorings and safe clique partitions, Congr. Numer. 181 (2006) 19-32. Zbl1113.05032
- [5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, 1998). Zbl0890.05002
- [6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs: Advanced Topics (Marcel Dekker, Inc., New York, 1998). Zbl0883.00011
- [7] O. Ore, Theory of Graphs (Amer. Math. Soc. Colloq. Publ. 38, 1962).
- [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. Zbl0803.05049

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