Proof of a conjecture on image-tuple domination in graphs Original Research Article

Let G=(V,E)G=(V,E) be a graph and NG[v]NG[v] the closed neighborhood of a vertex vv in GG. For k∈Nk∈N, the minimum cardinality of a set D⊆VD⊆V with |NG[v]∩D|≥k|NG[v]∩D|≥k for all v∈Vv∈V is the kk-tuple domination number γ×k(G)γ×k(G) of GG. In this note we prove the following conjecture of Rautenbach and Volkmann [D. Rautenbach, L. Volkmann, New bounds on the kk-domination number and the kk-tuple domination number, Appl. Math. Lett. 20 (2007) 98–102]: If k∈Nk∈N and G=(V,E)G=(V,E) is a graph of order nn and minimum degree δ≥kδ≥k, then View the MathML sourceγ×k(G)≤nδ+2−k(ln(δ+2−k)+ln(∑v∈V(dG(v)+1k−1))−ln(n)+1). Turn MathJax on