Signed total $k$-domatic numbers of digraphs

Maryam Atapour, Seyed Mahmoud Sheikholeslami, Lutz Volkmann

Let $D$ be a finite and simple digraph with vertex set $V(D)$, and let $f:V(D)\rightarrow\{-1,1\}$ be a two-valued function. If $k\ge 1$ is an integer and $\sum_{x\in N^-(v)}f(x)\ge k$ for each $vn V(D)$, where $N^-(v)$ consists of all vertices of $D$ from which arcs go into $v$, then $f$ is a signed total $k$-dominating function on $D$. A set $\{f_1,f_2,\ldots,f_d\}$ of distinct signed total $k$-dominating functions of $D$ with the property that $\sum_{i=1}^df_i(v)\le 1$, for each $v\in V(D)$, is called a {\em signed total $k$-dominating family} (of functions) of $D$. The maximum number of functions in a signed total $k$-dominating family of $D$ is the {\em signed total $k$-domatic number} of $D$, denoted by $d^t_{kS}(D)$. In this note we initiate the study of the signed total $k$-domatic numbers of digraphs and present some sharp upper bounds for this parameter.