Signed Distance \(k\)-domatic Numbers of Graphs

Let \( k \) be a positive integer and let \( G \) be a simple graph with vertex set \( V(G) \). If \( v \) is a vertex of \( G \), then the open \( k \)-neighborhood of \( v \), denoted by \( N_{k,G}(v) \), is the set \( N_{k,G}(v) = \{u \mid u \neq v \text{ and } d(u, v) \leq k\} \). The closed \( k \)-neighborhood of \( v \), denoted by \( N_{k,G}[v] \), is \( N_{k,G}[v] = N_{k,G}(v) \cup \{v\} \). A function \( f: V(G) \to \{-1,1\} \) is called a \emph{signed distance \( k \)-dominating function} if \( \sum_{u \in N_{k,G}(v)} f(u) \geq 1 \) for each vertex \( v \in V(G) \). A set \( \{f_1, f_2, \ldots, f_d\} \) of signed distance \( k \)-dominating functions on \( G \) with the property that \( \sum_{i=1}^d f_i(v) \leq 1 \) for each \( v \in V(G) \) is called a \emph{signed distance \( k \)-dominating family} (of functions) on \( G \). The maximum number of functions in a signed distance \( k \)-dominating family on \( G \) is the \emph{signed distance \( k \)-domatic number} of \( G \), denoted by \( d_{k,s}(G) \). Note that \( d_{1,s}(G) \) is the classical signed domatic number \( d_s(D) \). In this paper, we initiate the study of signed distance \( k \)-domatic numbers in graphs and we present some sharp upper bounds for \( d_{k,s}(G) \).