**Title:** A Note on Dominator Chromatic Number of Graphs

**Montes Taurus J. Pure Appl. Math.** / ISSN: 2687-4814

**Article ID:** MTJPAM-D-21-00012; **Volume 3 / Issue 2 / Year 2021**, Pages 1-7

**Document Type:** Research Paper

**Author(s):** Raghavachar Rangarajan ^{a} , David Ashok Kalarkop ^{b}

^{a}Department of Studies in Mathematics, University of Mysore, Manasagangothri, Mysuru – 570 006, India

^{b}Department of Studies in Mathematics, University of Mysore, Manasagangothri, Mysuru – 570 006, India

Received: 8 January 2021, Accepted: 4 March 2021, Published: 31 March 2021.

**Corresponding Author:** David Ashok Kalarkop (Email address: david.ak123@gmail.com)

**Full Text:** PDF

**Abstract**

There are a lot of interesting results on dominator chromatic number *χ*_{d}(*G*) of a graph *G*, along with certain open problems in the recent literature. In this paper, an attempt is made to answer partially to some of the open problems. We study the effect on dominator chromatic number of graphs on removal (addition) of some edges in some classes of graphs. We also exhibit some non-isomorphic graphs having same dominator chromatic number and present some graphs such that *χ*_{d}(*G*)=*χ*(*G*), *χ*_{d}(*G*)=*γ*(*G*) and *χ*_{d}(*μ*(*G*)) = *χ*_{d}(*G*)+2, where *μ*(*G*) is the Mycielskian of the graph *G*.

**Keywords:** Dominator coloring, Dominator chromatic number, Domination number

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