On the total \(\{ k\} \)-domination and total \(\{ k\} \)-domatic number of graphs
A total \(\{k\}\)-dominating function of a graph \(G\) (without isolated vertices) is a function \(f : V(G) \to \{0,1,\dots,k\}\) such that \(\sum_{\{u\mid uv\in E(G)\}}f(u)\geq k\) for all \(v\in V(G)\); the weight of \(f\) is \(\omega(f) = \sum_{v\in V(G)}f(v)\). The total \(\{k\}\)-domination number, \(\gamma_t^{\{k\}}(G)\), is the minimum weight of all such functions \(f\). A set \(\{f_1,f_2,\dots,f_d\}\) of \(k\) distinct such functions such that \(\sum_{i=1}^df_i(v)\leq k\) for all \(v\in V(G)\) is a total \(\{k\}\)-dominating family on \(G\), whose maximum number of members is the total \(\{k\}\)-domatic number, \(d_t^{\{k\}}(G)\) [\textit{S. M. Sheikholeslami} and \textit{L. Volkmann}, J.\ Comb.\ Optim.\ 23, No.\ 2, 252--260 (2012; Zbl 1243.90229)]. The Cartesian product of a cycle and a path is called a cylinder. From the authors' abstract: ``In this paper we present bounds for the total \(\{k\}\)-domination number and total \(\{k\}\)-domatic number. In addition, we determine the total \(\{k\}\)-domatic number of cylinders, and we give a Nordhaus-Gaddum type result.''
scientific article; zbMATH DE number 6143133