IJPAM: Volume 120, No. 4 (2018)

Title

COLOR CLASS DOMINATION AND
COLORFUL DOMINATION

Authors

V. Praba$^1$, V. Swaminathan$^2$, P. Aristotle$^3$
$^1$Sri Chandrasekharendra Saraswathi Viswa Mahavidyalaya
Kanchipuram, 631 561, Tamilnadu, INDIA
$^1$Department of Mathematics
Rajalakshmi Engineering College
Chennai, 602 105, Tamilnadu, INDIA
$^2$Ramanujan Research Center in Mathematics
Saraswathi Narayanan College
Madurai, 625 022, Tamilnadu, INDIA
$^3$PG & Research Department of Mathematics
Raja Doraisingam Government Arts College
Sivagangai, 630 561, Tamilnadu, INDIA

Abstract

Let $G=(V, E)$ be a finite, simple and undirected graph. A partition of $V(G)$ into independent sets such that each independent set is dominated by a vertex belonging to $V$ is called a color class domination partition (in short $cd$-partition) [[3], [4], [10], [11], [12]]. The minimum cardinality of a $cd$-partition is called the $cd$-chromatic number of $G$ and is denoted by $\chi_{cd}(G)$. A proper coloring of $G$ in which each vertex of the graph dominates some color class is called a dominator coloring of $G$ and the minimum number of color classes in a dominator coloring of $G$ is called the dominator chromatic number of $G$ and is denoted by $\chi_{d}(G)$ [[1], [5],[6], [8]]. In a $\chi_{d}$-partition of $G$, any set formed by selecting one vertex each from every color class becomes a dominating set of $G$. That is, in such a dominating set each vertex has distinct color and no color is represented by more than one element. Thus, a $\chi_{d}$-coloring gives rise to a dominating set which can be rightly called a colorful dominating set of $G$ (also called gamma coloring of $G$ [8]). In the case of $\chi_{cd}$-partition, this does not happen generally. So, it is a nice problem to find the minimum cardinality of a $cd$-partition which will give rise to a colorful dominating set. This new parameter is denoted by $\chi_{\gamma}^{cd}(G)$. A study of this parameter is initiated in this paper.

History

Received: June 7, 2017
Revised: September 9, 2017
Published: August 4, 2019

AMS Classification, Key Words

AMS Subject Classification: 05C69
Key Words and Phrases: color class domination partition, dominator chromatic number, colorful dominating set, dominator coloring

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Bibliography

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How to Cite?

DOI: 10.12732/ijpam.v120i4.12 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2018
Volume: 120
Issue: 4
Pages: 623 - 638


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