Dominated chromatic number of some kinds of the generalized Helm graphs

Articles in Press

Document Type : Original paper

Authors

Department of Pure Mathematics and Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P. O. Box 1159-91775, Mashhad, Iran

Abstract

Let G be a simple graph. The dominated coloring of a graph G is a proper coloring of G such that each color class is dominated by at least one vertex. The minimum number of colors needed for a dominated coloring of G is called the dominated chromatic number of G, denoted by χdom(G). The current study is devoted to investigate the dominated chromatic number of Helm graphs and some  kinds of its the generalizations.

Keywords

Main Subjects

References

[1] S. Alikhani and M.R. Piri, On the dominated chromatic number of certain graphs, Trans. Comb. 9 (2020), no. 4, 217–230.  https://doi.org/10.22108/toc.2020.119361.1675
[2] S. Alikhani and M.R. Piri, Introduction to dominated edge chromatic number of a graph, Opuscula Math. 41 (2021), no. 2, 245–257.  https://doi.org/10.7494/OpMath.2021.41.2.245
[3] C. Berge, The Theory of Graphs, Courier Corporation, 2001.
[4] G. Chartrand and P. Zhang, Chromatic Graph Theory, Chapman and Hall/CRC, 2019.
[5] M. Chellali and L. Volkmann, Relations between the lower domination parameters and the chromatic number of a graph, Discrete Math. 274 (2004), no. 1-3, 1–8.  https://doi.org/10.1016/S0012-365X(03)00093-1
[6] Y.H. Chen, The dominated coloring problem and its application, International Conference on Computational Science and Its Applications (B. Murgante,  S. Misra, A.M.A. C. Rocha, C. Torre, J.G. Rocha, M.I. Falcäo, D. Taniar, B.O. Apduhan, and O. Gervasi, eds.), Springer, 2014, pp. 132–145.
[7] F. Choopani, A. Jafarzadeh, A. Erfanian, and D.A. Mojdeh, On dominated coloring of graphs and some Nordhaus–Gaddum-type relations, Turkish J. Math. 42 (2018), no. 5, 2148–2156.  https://doi.org/10.3906/mat-1710-97
[8] W. Imrich and S. Klavžar, Product Graphs: Structure and Recognition, Wiley, 2000.
[9] A.A. Khalil, Determination and testing the domination numbers of Helm graph, Web graph and Levi graph using matlab, J. Educ. Sci. 24 (1999), no. 2, 103–116.
[10] A.A. Khalil and O.A. Khalil, Determination and testing the domination numbers of tadpole graph, book graph and stacked book graph using MATLAB, College of Basic Education Researches Journal 10 (2010), no. 1, 491–506.
[11] H.B. Merouane, M. Haddad, M. Chellali, and H. Kheddouci, Dominated colorings of graphs, Graphs Combin. 31 (2015), no. 3, 713–727.  https://doi.org/10.1007/s00373-014-1407-3
[12] M. Nandi, S. Parui, and A. Adhikari, The domination numbers of cylindrical grid graphs, Appl. Math. Comput. 217 (2011), no. 10, 4879–4889.  https://doi.org/10.1016/j.amc.2010.11.019
[13] O. Ore, Theory of Graphs, American Mathematical Society Colloquium Publications, 1962.
[14] P. Pavlič and J. Žerovnik, Roman domination number of the Cartesian products of paths and cycles, Electron. J. Comb. 19 (2012), no. 3, Article Number: P19  https://doi.org/10.37236/2595
[15] P. Pavlič and J. Žerovnik, A note on the domination number of the Cartesian products of paths and cycles, Kragujevac J. Math. 37 (2013), no. 2, 275–285.
[16] F. Poryousefi, A. Erfanian, and M. Nasiri, Dominated chromatic number of semi-strong, strong and modular product of graphs, Submitted.
[17] F. Poryousefi, A. Erfanian, and M. Nasiri, Dominated chromatic number of some kinds of (n,n+1) graphs, Submitted.
[18] D.B. West, Introduction to Graph Theory, Prentice hall Upper Saddle River, 2001.
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 22 December 2024

Files

Share

How to cite

Statistics