Domination parameters of Cayley graphs of some groups

Subject Areas : Combinatorics, Graph theory

F. Ramezani ¹

1 - Department of Mathematics,Yazd University,Yazd, Iran

Received: 2020-01-06 Accepted : 2020-09-17 Published : 2020-09-01

Keywords: Cayley graph, ‎cyclic group‎, ‎domination number‎, ‎signed domination number, ‎cubic graph‎,

Abstract :

‎In this paper‎, ‎we investigate domination number‎, ‎$\gamma$‎, ‎as well‎‎as signed domination number‎, ‎$\gamma_{_S}$‎, ‎of all cubic Cayley‎‎graphs of cyclic and quaternion groups‎. ‎In addition‎, ‎we show that‎‎the domination and signed domination numbers of cubic graphs depend‎on each other‎.

References:

[1] A. Assari, M. Rahimi, Graphs generated by measures, J. of Mathematics. (2016), 2016:1706812.

[2] A. Badawi, On the Total Graph of a Ring and its Related Graphs: a survey, Commutative algebra, Springer, New York, 2014.

[3] T. T. Chelvam, G. Kalaimurugan, Bounds for Domination Parameters in Cayley Graphs on Dihedral Group, Open J. of Discrete Math. 2 (2012), 5-10.

[4] T. T. Chelvam, I. Rani, Dominating sets in Cayley graphs on Zn, Tamkang J. Math. 38 (4) (2007), 341-345.

[5] J. Dunbar, S. Hedetniemi, M. A. Henning, P. J. Slater, Signed domination in graphs, Graph theory, Combinatorics, and Algorithms, Wiley, New York, 1995.

[6] O. Favaron, Signed domination in regular graphs, Discrete Mathematics. 158 (1996), 287-293.

[7] R. Haas, T. B. Wexler, Bounds on the Signed Domination Number of a Graph, Electronic Notes in Discrete Mathematics, Elsevier, 2002.

[8] R. Haas, T. B. Wexler, Signed domination numbers of a graph and its complement, Discrete Mathematics. 283 (2004), 87-92.

[9] T. W. Haynes, S. T. Hedetniemi, P. J. Slater, Fundamentals of Domination in Graphs, Monographs and Textbooks in Pure and Applied Mathematics 208, 1998.

[10] M. A. Henning, P. J. Slater, Inequalities relating domination parameters in cubic graphs, Discrete Mathematics. 158 (1996), 87-98.

[11] C. H. Li, On isomorphisms of finite Cayley graphsa survey, Discrete Mathematics. 256 (2002), 301-334.

[12] A. Meir, J. W. Moon, Relations between packing and covering numbers of a tree, Pacific J. Math. 61 (1) (1975), 225-233.

[13] P. Pavlic, J. A. Zerovnik, Note on the domination number of the Cartesian products of paths and cycles, Krag. J. Math. 37 (2) (2013), 275-285.

[14] E. Vatandoos , F. Ramezani, Domination and signed domination number of Cayley graphs, IJMSI. 14 (1) (2019), 35-42.

[15] E. Vatandoost, F. Ramezani, On the domination and signed domination numbers of zero-divisor graph, EJGTA. 4 (2) (2016), 148-156.

[16] L. Volkmann, B. Zelinka, Signed domatic number of a graph, Discrete. Appl. mathematics. 150 (2005), 261-267.

[17] B. Zelinka, Signed and minus domination in bipartite graphs, Czechoslovak Math. 56 (131) (2006), 587-590.

[18] B. Zelinka, Some remarks on domination in cubic graphs, Discrete Mathematics. 158 (1996), 249-255.