The total {k}-domatic number of a graph

Abstract

For a positive integer k, a total {k}-dominating function of a graph G is a function f from the vertex set V(G) to the set {0,1,2,…,k} such that for any vertex v∈V(G), the condition ∑ _u∈N(v) f(u)≥k is fulfilled, where N(v) is the open neighborhood of v. A set {f ₁,f ₂,…,f _d } of total {k}-dominating functions on G with the property that $\sum_{i=1}^{d}f_{i}(v)\le k$ for each v∈V(G), is called a total {k}-dominating family (of functions) on G. The maximum number of functions in a total {k}-dominating family on G is the total {k}-domatic number of G, denoted by $d_{t}^{\{k\}}(G)$. Note that $d_{t}^{\{1\}}(G)$ is the classic total domatic number d _t (G). In this paper we initiate the study of the total {k}-domatic number in graphs and we present some bounds for $d_{t}^{\{k\}}(G)$. Many of the known bounds of d _t (G) are immediate consequences of our results.