Abstract
We continue the study of $\{k\}$-dominating functions in graphs (or integer domination as we shall also say) started by Domke, Hedetniemi, Laskar, and Fricke~[5]. For $k \ge 1$ an integer, a function $f \colon V(G) \rightarrow \{0,1,\ldots,k\}$ defined on the vertices of a graph $G$ is called a $\{k\}$-dominating function if the sum of its function values over any closed neighborhood is at least~$k$. The weight of a $\{k\}$-dominating function is the sum of its function values over all vertices. The $\{k\}$-domination number of $G$ is the minimum weight of a $\{k\}$-dominating function of $G$. We study the $\{k\}$-domination number on the Cartesian product of graphs, mostly on problems related to the famous Vizing's conjecture. A connection between the $\{k\}$-domination number and other domination type parameters is also studied.
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