Year: 2026
Type: article
Abstract: A k -rainbow total dominating function of a graph G is a function $$f: V(G) \rightarrow 2^{[k]}$$ f : V ( G ) → 2 [ k ] such that for every vertex $$v \in V(G)$$ v ∈ V ( G ) with $$f(v) = \emptyset $$ f ( v ) = ∅ , the union of the colors assigned to its neighbors equals [ k ], and if $$f(v) = \{i\}$$ f ( v ) = { i } , then v has a neighbor u with $$i \in f(u)$$ i ∈ f ( u ) . The minimum weight of such a function is called the k -rainbow total domination number of G and is denoted by $$\gamma _{k\textrm{rt}}(G)$$ 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 $$\gamma _{k\textrm{rt}}(G)$$ 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 $$\gamma _{k\textrm{rt}}(G)/\gamma (G)$$ 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 $$\gamma _{k\textrm{rt}}(G)$$ γ k rt
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
Award: Graph theory
OpenAlex ID: W7130513340