Article ID: MTJPAM-D-21-00012

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

aDepartment of Studies in Mathematics, University of Mysore, Manasagangothri, Mysuru – 570 006, India

bDepartment 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 χd(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 χd(G)=χ(G), χd(G)=γ(G) and χd(μ(G)) = χd(G)+2, where μ(G) is the Mycielskian of the graph G.

Keywords: Dominator coloring, Dominator chromatic number, Domination number

References:
  1. S. Arumugam, J. Bagga and K. R. Chandrasekar, On dominator coloring in graphs, Proc. Indian Acad. Sci. (Math. Sci.) 122 (4), 561-571, 2012.
  2. R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, Springer, 2012.
  3. J. A. Bondy and U. S. R. Murty, Graph Theory, Springer, 2008.
  4. S. Brandt, Triangle-free graphs whose independence number equals the degree, Discrete Math. 310 (3), 662-669, 2010.
  5. G. Chartrand and L. Lesniak, Graphs and Digraphs, (4th edition) Chapman and Hall, CRC, 2011.
  6. X. Chen and H. Xing, Domination parameters in Mycielski graphs, Util. Math. 71, 235-244, 2006.
  7. R. Gera, On dominator coloring in graphs, Graph Theory Notes N. Y. 52, 25-30, 2007.
  8. R. Gera, C. Rasmussen and S. Horton, Dominator colorings and safe clique partitions, Congr. Numer. 181, 19-32, 2006.
  9. F. Harary, Graph Theory, Narosa Publishing House, 2005.
  10. T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc., New York, 1998.
  11. T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Domination in Graphs-Advanced Topics, Marcel Dekker, Inc., New York, 1998.
  12. A. Mohammed Abid and T. R. Ramesh Rao, Dominator coloring of Mycielskian graphs, Australas. J. Combin. 73(2), 274-279, 2019.
  13. J. Mycielski, Sur le colouriage des graphes, Colloq. Math. 3, 161-162, 1955.