# IJPAM: Volume 120, No. 4 (2018)

# Title

COLOR CLASS DOMINATION ANDCOLORFUL DOMINATION

# Authors

V. Praba, V. Swaminathan, P. AristotleSri Chandrasekharendra Saraswathi Viswa Mahavidyalaya

Kanchipuram, 631 561, Tamilnadu, INDIA

Department of Mathematics

Rajalakshmi Engineering College

Chennai, 602 105, Tamilnadu, INDIA

Ramanujan Research Center in Mathematics

Saraswathi Narayanan College

Madurai, 625 022, Tamilnadu, INDIA

PG & Research Department of Mathematics

Raja Doraisingam Government Arts College

Sivagangai, 630 561, Tamilnadu, INDIA

# Abstract

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

**DOI: 10.12732/ijpam.v120i4.12**

International Journal of Pure and Applied Mathematics

**How to cite this paper?****Source:****ISSN printed version:**1311-8080

**ISSN on-line version:**1314-3395

**Year:**2018

**Volume:**120

**Issue:**4

**Pages:**623 - 638

Google Scholar; DOI (International DOI Foundation); WorldCAT.

**This work is licensed under the Creative Commons Attribution International License (CC BY).**