On the \(\{k\}\)-domatic number of graphs (Q2831595)

\(\{ k \}\)-domatic numbers of graphs are defined as a generalization of domatic numbers. Let \(G=(V,E)\) be a graph. A \(\{ k \}\)-dominating function is a function \(f:(V) \to \{0,1,\dots,k \}\) such that the summation of values of \(f\) over any closed neighborhood of a vertex is at least \(k\). A set \(\{ f_1,f_2,\dots,f_d\} \) of \(\{ k \}\)-dominating functions with property that \(\sum_{i=1}^{d}f_i(v)\leq k \) for each \(v\in V\), is called a \(\{ k \}\)-dominating family. The maximum number of functions in a \(\{ k \}\)-dominating family is the \(\{ k \}\)-domatic number.NEWLINENEWLINEBasic bounds of \(\{ k \}\)-domatic numbers involving minimal degree of graph \(G\) are presented. The authors then find exact values of \(\{ k \}\)-domatic numbers for cylinders. A cylinder is defined as Cartesian product of a cycle and path.