Journal of the
Korean Mathematical Society JKMS

Let $k$ be a positive integer and let $G$ be a simple graph with vertex set $V(G)$. A function $f:V(G)\longrightarrow \{-1,1\}$ is called a signed total $k$-dominating function if $\sum_{u\in N(v)}f(u)\ge k$ for each vertex $v\in V(G)$. A set $\{f_1,f_2,\ldots,f_d\}$ of signed total $k$-dominating functions of $G$ with the property that $\sum_{i=1}^df_i(v)\le 1$, for each $v\in V(G)$, is called a signed total $k$-dominating family (of functions) of $G$. The maximum number of functions in a signed total $k$-dominating family of $G$ is the signed total $k$-domatic number of $G$, denoted by $d^t_{kS}(G)$. In this note we initiate the study of the signed total $k$-domatic numbers of graphs and present some sharp upper bounds for this parameter. We also determine the signed total $k$-domatic numbers of complete graphs and complete bipartite graphs.

Keywords: signed total $k$-domatic number, signed total $k$-dominating function, signed total $k$-domination number

MSC numbers: 05C69