On k-rainbow total domination and a related conjectureErveš, Rija (Avtor) Kraner Šumenjak, Tadeja (Avtor) Tepeh, Aleksandra (Avtor) dominaitonrainbow total dominationNP-completeA k-rainbow total dominating function of a graph G is a function f : V(G) → 2[k] such that for every vertex v ∈ V(G) with f (v) = ∅, the union of the colors assigned to its neighbors equals [k], and if f (v) = {i}, then v has a neighbor u with i ∈ f (u). The minimum weight of such a function is called the k-rainbow total domination number of G and is denoted by γkrt(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 γkrt(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 γkrt(G)/γ (G), originally established by Ojakian et al. (2021), but also contributes to disproving their conjecture on the lower bound for γkrt(G) when k = 4.Springer20262026-02-24 07:46:13Članek v reviji97220sl© The Author(s) 2026