Signed {k}-domatic numbers of graphs

Let \( k \) be a positive integer, and let \( G \) be a simple graph with vertex set \( V(G) \). A function \( f: V(G) \to \{\pm1, \pm2, \ldots, \pm k\} \) is called a signed \(\{k\}\)-dominating function if

\[ \sum_{u \in N[v]} f(u) \geq k \]

for each vertex \( v \in V(G) \).

The signed \(\{1\}\)-dominating function is the same as the ordinary signed domination. A set \( \{f_1, f_2, \ldots, f_d\} \) of signed \(\{k\}\)-dominating functions on \( G \) with the property that

\[ \sum_{i=1}^d f_i(v) \leq k \]

for each \( v \in V(G) \), is called a \emph{signed \(\{k\}\)-dominating family} (of functions) on \( G \). The maximum number of functions in a signed \(\{k\}\)-dominating family on \( G \) is the \emph{signed \(\{k\}\)-domatic number} of \( G \), denoted by \( d_{\{k\}S}(G) \). Note that \( d_{\{1\}S}(G) \) is the classical signed domatic number \( d_s(G) \).

In this paper, we initiate the study of signed \(\{k\}\)-domatic numbers in graphs, and we present some sharp upper bounds for \( d_{\{k\}S}(G) \). In addition, we determine \( d_{\{k\}S}(G) \) for several classes of graphs. Some of our results are extensions of known properties of the signed domatic number.