Opuscula Math. 41, no. 2 (2021), 245-257
https://doi.org/10.7494/OpMath.2021.41.2.245

Opuscula Mathematica

# Introduction to dominated edge chromatic number of a graph

Saeid Alikhani

Abstract. We introduce and study the dominated edge coloring of a graph. A dominated edge coloring of a graph $$G$$, is a proper edge coloring of $$G$$ such that each color class is dominated by at least one edge of $$G$$. The minimum number of colors among all dominated edge coloring is called the dominated edge chromatic number, denoted by $$\chi_{dom}^{\prime}(G)$$. We obtain some properties of $$\chi_{dom}^{\prime}(G)$$ and compute it for specific graphs. Also examine the effects on $$\chi_{dom}^{\prime}(G)$$, when $$G$$ is modified by operations on vertex and edge of $$G$$. Finally, we consider the $$k$$-subdivision of $$G$$ and study the dominated edge chromatic number of these kind of graphs.

Keywords: dominated edge chromatic number, subdivision, operation, corona.

Mathematics Subject Classification: 05C25.

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1. S. Alikhani, E. Deutsch, More on domination polynomial and domination root, Ars Combin. 134 (2017), 215-232.
2. S. Alikhani, M.R. Piri, Dominated chromatic number of some operations on a graph, arXiv:1912.00016 [math.CO].
3. S. Alikhani, S. Soltani, Distinguishing number and distinguishing index of neighbourhood coronal of two graphs, Contrib. Discrete Math. 14 (2019), no. 1, 175-189.
4. S. Alikhani, S. Soltani, Distinguishing number and distinguishing index of natural and fractional powers of graphs, Bull. Iran. Math. Soc. 43 (2017), no. 7, 2471-2482.
5. S. Alikhani, S. Soltani, Trees with distinguishing number two, AKCE International J. Graphs Combin. 16 (2019), 280-283.
6. F. Choopani, A. Jafarzadeh, D.A. Mojdeh, On dominated coloring of graphs and some Nardhaus-Gaddum-type relations, Turkish J. Math. 42 (2018), 2148-2156.
7. R. Gera, S. Horton, C. Ramussen, Dominator colorings and safe clique partitions, Conress. Num. 181 (2006), 19-32.
8. N. Ghanbari, S. Alikhani, More on the total dominator chromatic number of a graph, J. Inform. Optimiz. Sci. 40 (2019), no. 1, 157-169.
9. N. Ghanbari, S. Alikhani, Total dominator chromatic number of some operations on a graph, Bull. Comp. Appl. Math. 6 (2018), no. 2, 9-20.
10. N. Ghanbari, S. Alikhani, Introduction to total dominator edge chromatic number, TWMS J. App. Eng. Math. (to appear).
11. B. Grunbaum, Acyclic coloring of planar graphs, Israel J. Math. 14 (1973), 390-408.
12. A.V. Kostochka, M. Mydlarz, E. Szemeredi, H.A. Kierstead, A fast algorithm for equitable coloring, Combinatorica 30 (2010), no. 2, 217-224.
13. V.R. Kulli, D.K. Patwari, On the total edge domination number of graph, [in:] A.M. Mathi (ed.), Proc. of the Symp. on Graph Theory and Combinatorics, Kochi Centre Math. Sci, Trivandrum, Series Publication 21 (1991), 75-81.
14. H.B. Merouane, M. Chellali, M. Haddad, H. Kheddouci, Dominated coloring of graphs, Graphs Combin. 31 (2015), 713-727.
15. M. Walsh, The hub number of a graph, Int. J. Math. Comput. Sci. 1 (2006), 117-124.
16. X. Zhu, Circular chromatic number: a survey, Discrete Math. 229 (2001), 371-410.
17. X. Zhou, T. Nishizeki, S. Nakano, Edge-coloring algorithms,, Technical report, Graduate School of Information Sciences, Tohoku University, Sendai 980-77, Japan, 1996.
• Yazd University, Department of Mathematics, 89195-741, Yazd, Iran
• Communicated by Dalibor Fronček.
• Revised: 2020-12-24.
• Accepted: 2021-02-04.
• Published online: 2021-03-17.