IJPAM: Volume 104, No. 2 (2015)

EQUITABLE DOMINATING CHROMATIC SETS IN GRAPHS

L. Muthusubramanian$^1$, S.P. Subbiah$^2$, V. Swaminathan$^3$
$^1$Department of Mathematics
Sethu Institute of Technology
Kariapatti, Tamilnadu, INDIA
$^2$Department of Mathematics
Mannar Thirumalai Naicker College
Madurai, Tamilnadu, INDIA
$^3$Ramanujan Research Center in Mathematics
Saraswathi Narayanan College
Madurai, Tamilnadu, INDIA


Abstract. Let $G=(V,E)$ be a simple graph. A subset $D$ of $V(G)$ is said to be an equitable dominating set of $G$ if for every vertex $v\in V-D$ there exists a vertex $u\in D$ such that $uv\in E(G)$ and $\vert d(u)-d(v)\vert\leq 1$. A subset $D$ of $V(G)$ is said to be an equitable dominating chromatic set of $G$ if $D$ is an equitable dominating set of $G$ and $\chi(<D>)=\chi(G)$. Since $V$ is an equitable dominating chromatic set of $G$, the existence of equitable dominating chromatic set in a graph is guaranteed. The minimum cardinality of such a set is called the equitable dominating chromatic number of $G$ and is denoted by $\gamma_{ch}^e(G)$. The property of equitable dominating chromatic set is super hereditary. Hence equitable dominating chromatic set is minimal if and only if it is $1$-minimal. Characterization of minimal equitable dominating chromatic sets is derived. The values of $\gamma_{ch}^e(G)$ for many classes of graphs have been found. It is established that $1\leq \gamma_{ch}^e(G)\leq n$. Interesting results are proved with respect to the new parameters.

Received: April 29, 2015

AMS Subject Classification: 05C17, 05C69, 05C70

Key Words and Phrases: equitable dominating set, equitable dominating chromatic set

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DOI: 10.12732/ijpam.v104i2.4 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: 2015
Volume: 104
Issue: 2
Pages: 193 - 202


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