On k-Rainbow Total Domination and a Related Conjecture

A k -rainbow total dominating function of a graph G is a function f: V(G) → 2[k] f : V ( G ) → 2 [ k ] such that for every vertex v ∈ V(G) v ∈ V ( G ) with f(v) = ∅ f ( v ) = ∅ , the union of the colors assigned to its neighbors equals [ k ], and if f(v) = \ f ( v ) = { i } , then v has a neighbor u with i ∈ f(u) i ∈ f ( u ) . The minimum weight of such a function is called the k -rainbow total domination number of G and is denoted by γ ₖᵣₜ(G) γ k 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 γ ₖᵣₜ(G) γ k 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 γ ₖᵣₜ(G)/γ (G) γ k rt ( G ) / γ ( G ) , originally established by Ojakian et al. (2021), but also contributes to disproving their conjecture on the lower bound for γ ₖᵣₜ(G) γ k rt