×

The Italian domatic number of a digraph. (English) Zbl 1412.05085

Summary: An Italian dominating function on a digraph \(D\) with vertex set \(V(D)\) is defined as a function \(f:V(D)\to \{0, 1, 2\}\) such that every vertex \(v\in V(D)\) with \(f(v)=0\) has at least two in-neighbors assigned 1 under \(f\) or one in-neighbor \(w\) with \(f(w)=2\). A set \(\{f_1,f_2,\dots,f_d\}\) of distinct Italian dominating functions on \(D\) with the property that \(\sum_{i=1}^d f_i(v)\leq 2\) for each \(v\in V(D)\), is called an Italian dominating family (of functions) on \(D\). The maximum number of functions in an Italian dominating family on \(D\) is the Italian domatic number of \(D\), denoted by \(d_{I}(D)\). In this paper, we initiate the study of the Italian domatic number in digraphs, and we present some sharp bounds for \(d_{I}(D)\). In addition, we determine the Italian domatic number of some digraphs.

MSC:

05C20 Directed graphs (digraphs), tournaments
05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
PDFBibTeX XMLCite