A k-rainbow total dominating function of a graph G is a function \(f: V(G) \rightarrow 2^{[k]}\) such that for every vertex \(v \in V(G)\) with \(f(v) = \emptyset \) , the union of the colors assigned to its neighbors equals [k], and if \(f(v) = \{i\}\) , then v has a neighbor u with \(i \in f(u)\) . The minimum weight of such a function is called the k-rainbow total domination number of G and is denoted by \(\gamma _{k\textrm{rt}}(G)\) . We contribute to the study of k-rainbow total domination by proving one conjecture and constructing a counterexample to another. First, we show that the problem of determining whether a graph admits a k-rainbow total dominating function of a given weight is NP-complete. In the second part, we derive an upper bound on the domination number of a graph G in terms of \(\gamma _{k\textrm{rt}}(G)\) and the frequency of the least-used color in a k-rainbow total dominating function. This result not only provides an alternative and shorter proof of a known lower bound on \(\gamma _{k\textrm{rt}}(G)/\gamma (G)\) , originally established by Ojakian et al. (2021), but also contributes to disproving their conjecture on the lower bound for \(\gamma _{k\textrm{rt}}(G)\) when \(k= 4\) .
