The most famous open problem involving domination in graphs is Vizing’s conjecture which states that the domination number of the Cartesian product of any two graphs is at least as large as the product of their domination numbers. In this paper, we investigate a similar problem for global equitable domination. In particular, we explore the multiplicative nature of the global equitable domination number  on the Cartesian products 

global dominating set, equitable dominating set, global equitable dominating set, global equitable domination number, Cartesian product.

Received: April 19, 2024; Revised: May 10, 2024; Accepted: May 29, 2024

How to cite this article: S. K. Vaidya and R. M. Pandit, Global equitable domination in Cartesian product of graphs, Advances and Applications in Discrete Mathematics 41(5) (2024), 341-356. https://doi.org/10.17654/0974165824025

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References:
[1] R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, 2nd ed., Springer, New York, 2012.
[2] B. Basavanagoud and V. V. Teli, Equitable global domination in graphs, International Journal of Mathematical Archive 6(3) (2015), 122-125.
[3] B. Hartnell and D. F. Rall, Domination in Cartesian products: Vizing’s conjecture, Domination in graphs, Vol. 209 of Monogr. Textbooks Pure Appl. Math. Dekker, New York, 1998, pp. 163-189.
[4] B. L. Hartnell and D. F. Rall, Improving some bounds for dominating Cartesian products, Discuss. Math. Graph Theory 23(2) (2003), 261-272.
[5] T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of domination in graphs, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, 1998.
[6] T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Domination in graphs - advanced topics, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, 1998.
[7] X. Hou and Y. Lu, On the -domination number of Cartesian products of graphs, Discrete Math. 309 (2009), 3413-3419.
[8] V. R. Kulli and B. Janakiram, The total global domination number of a graph, Indian J. Pure Appl. Math. 27(6) (1996), 537-542.
[9] D. F. Rall, Packing and domination invariants on Cartesian products and direct products, Pre-conference proceedings of International Conference on Discrete Mathematics, Bangalore, India, 2006.
[10] E. Sandueta, Equitable domination in some graphs, Appl. Math. Sci. 13(7) (2019), 309-314.
[11] E. Sampathkumar, The global domination number of a graph, Journal of Mathematical and Physical Sciences 23(5) (1989), 377-385.
[12] V. Swaminathan and K. Dharmalingam, Degree equitable domination on graphs, Kragujevac J. Math. 35(1) (2011), 191-197.
[13] S. K. Vaidya and R. M. Pandit, Some new results on global dominating sets, ISRN Discrete Mathematics, Vol. 2012, Article ID 852129, 6 pages, 2012.
doi: 10.5402/2012/852129.
[14] S. K. Vaidya and R. M. Pandit, Some results on global dominating sets, Proyecciones 32(3) (2013), 235-244.
[15] S. K. Vaidya and R. M. Pandit, Global equitable domination number of some wheel related graphs, International Journal of Mathematical and Combinatorics 3 (2016), 77-85.
[16] S. K. Vaidya and R. M. Pandit, Global equitable domination in some degree splitting graphs, Notes on Number Theory and Discrete Mathematics 24(2) (2018), 74-84.
[17] S. K. Vaidya and R. M. Pandit, The global equitable domination in graphs, Advances and Applications in Discrete Mathematics 39(2) (2023), 155-167.
[18] V. G. Vizing, Some unsolved problems in graph theory, Uspekhi Mat. Nauk 23(6(144)) (1968), 117-134.
[19] D. B. West, Introduction to Graph Theory, Prentice Hall of India, New Delhi, 2003.