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Upper Bounds on the k -Tuple (Roman) Domination Number of a Graph
Graphs and Combinatorics ( IF 0.6 ) Pub Date : 2020-11-03 , DOI: 10.1007/s00373-020-02249-7
Michael A. Henning , Nader Jafari Rad

Rautenbach and Volkmann (Appl Math Lett 20:98–102, 2007), gave an upper bound for the k-tuple domination number of a graph. Rad (J Combin Math Comb Comput, 2019, in press) presented an improvement of the above bound using the Caro-Wei Theorem. In this paper, using the well-known Brooks’ Theorem for vertex coloring and vertex covers, we improve the above bounds on the k-tuple domination number under some certain conditions. In the special case \(k=1\), we improve the upper bounds for the domination number (Arnautov in Prikl Mat Program 11:3–8, 1974; Payan in Cahiers Centre Études Recherche Opér 17:307–317, 1975) and the Roman domination number (Cockayne et al. in Discrete Math 278:11–22, 2004). We also improve bounds given by Hansberg and Volkmann (Discrete Appl Math 157:1634–1639, 2009) for Roman k-domination number, and Rad and Rahbani (Discuss Math Graph Theory 39:41–53, 2019) for double Roman domination number.



中文翻译:

图的k-元组(罗马)支配数的上界

Rautenbach和Volkmann(Appl Math Lett 20:98–102,2007)给出了图的k元组控制数的上限。Rad(J Combin Math Comb Comput,2019年出版)使用Caro-Wei定理对上述界限进行了改进。在本文中,使用著名的布鲁克斯定理进行顶点着色和顶点覆盖,我们在某些条件下改善了k元组控制数的上述界限。在特殊情况下\(k = 1 \),我们提高了支配数的上限(1974年,Prikl Mat计划中的Arnautov; 1974年; Cahiers CenterÉtudesRechercheOpér:17:307-317,Payan; 1975年)和罗马支配数(Cockayne等。离散数学278:11-22,2004年)。我们还改善了Hansberg和Volkmann(Discrete Appl Math 157:1634-1639,2009 )对于罗马k支配数以及Rad和Rahbani(Discuss Math Graph Theory 39:41–53,2019)对于双罗马支配数的界限。 。

更新日期:2020-11-04
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