Dominated chromatic number of some kinds of the generalized Helm graphs

[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.