Global Dominator Chromatic Number of Certain Graphs

Document Type : Original Scientific Paper

Authors

‎Department of Mathematical Sciences,‎ ‎Yazd University, ‎Yazd‎, ‎Iran

10.22052/mir.2025.255889.1487

Abstract

‎For a graph G=(V,E) and a vertex subset $D\subseteq V$‎, ‎a vertex $v\in V$ is called a dominator of D if v is adjacent to every vertex in D‎, ‎and an anti-dominator of D if v is not adjacent to any vertex in D. ‎Given a coloring $C=\{V_{1},V_{2},\ldots,V_{k}\}$ of $G$‎, ‎a color {class $V_{i}$} {is a dominating color class (resp. an anti dominating color class) for a vertex ‎v‎‎ if ‎‎v‎‎ dominates all vertices in ‎$‎V_i‎$‎ (resp. ‎‎v‎‎ dominates no vertex in ‎$‎V_i‎$‎)}‎. ‎A coloring C is a global dominator coloring if each vertex in $G$ has both a dominating and an anti-dominating color class‎. ‎The global dominator chromatic number‎, ‎denoted by $\chi_{gd}(G)$‎, ‎is the minimum number of colors required for a global dominator coloring of $G$‎. ‎In this paper‎, ‎we investigate the global dominator chromatic number for various classes of graphs‎.

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References

[1] R. Gera, C. Rasmussen and S. Horton, Dominator colorings and safe clique partitions, Congr. Numer. 181 (2006) 19 - 32.
[2] S. Alikhani, N. Ghanbari and S. Soltani, Total dominator chromatic number of k-subdivision of graphs, Art Discrete Appl. Math. 6 (2023) #1.10, https://doi.org/10.26493/2590-9770.1495.2a1.
[3] H. B. Merouane and M. Chellali, On the dominator colorings in trees, Discuss. Math. Graph Theory 32 (2012) 677 - 683.
[4] I. S. Hamid and M. Rajeswari, Global dominator coloring of graphs, Discuss. Math. Graph Theory 39 (2019) 325-339, https://doi.org/10.7151/dmgt.2089.
[5] R. Rangarajan and D. A. Kalarkop, A note on global dominator coloring of graphs, Discrete Math. Algorithms Appl. 14 (2022) #2150158, https://doi.org/10.1142/S1793830921501585.
[6] S. Askari, D. A. Mojdeh and E. Nazari, Total global dominator chromatic number of graphs, TWMS J. App. and Eng. Math. 12 (2022) 650 - 661.
[7] T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Domination in Graphs: Advanced Topics, Marcel Dekker, New York 1998.
[8] S. Arumugam and R. Kala, A note on global domination in graphs, Ars Combin. 93 (2009) 175 - 180.
[9] F. Harary and G. E. Uhlenbeck, On the number of Husimi trees, I. Proc. Nat. Acad. Sci. U.S.A. 39 (1953) 315 - 322.
[10] K. Husimi, Note on Mayer’s theory of cluster integrals, J. Chem. Phys. 18 (1950) 682 - 684, https://doi.org/10.1063/1.1747725.
[11] R. J. Riddell, Contributions to the theory of condensation, Ph.D. Thesis, Univ. of Michigan, Ann Arbor, 1951.
[12] M. Chellali, Bounds on the 2-domination number in cactus graphs, Opuscula Math. 26 (2006) 5 - 12.
[13] N. Ghanbari and S. Alikhani, Sombor index of certain graphs, Iranian J. Math. Chem. 12 (2021) 27 - 37,
https://doi.org/10.22052/IJMC.2021.242106.1547.
[14] S. Majstorovic, T. Došlic and A. Klobucar, k-domination on hexagonal cactus chains, Kragujevac J. Math. 36 (2012) 335 - 347.
[15] S. Alikhani, S. Jahari, M. Mehryar and R. Hasni, Counting the number of dominating sets of cactus chains, Opt. Adv. Mat. Rapid Comm. 8 (2014) 955 - 960.
[16] S. Alikhani, H. R. Golmohammadi and E. V. Konstantinova, Coalition of cubic graphs of order at most 10, Commun. Comb. Optim. 9 (2024) 437-450.
[17] S. Alikhani and Y. H. Peng, Domination polynomials of cubic graphs of order 10, Turkish J. Math. 35 (2011) 355-366, https://doi.org/10.3906/mat-1002-141.
Mathematics Interdisciplinary Research
Volume 10, Issue 2
In Progress
June 2025
Pages 183-198

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