New bounds on the kk-domination number and the kk-tuple domination number

Let G=(V,E)G=(V,E) be a graph, k∈Nk∈N and let NG(v)NG(v) and dG(v)dG(v) denote the neighbourhood and degree of a vertex v∈Vv∈V in GG, respectively. The minimum cardinality of a set D⊆VD⊆V with |NG(v)∩D|≥k|NG(v)∩D|≥k for all v∈V∖Dv∈V∖D is the kk-domination number γk(G)γk(G) of GG. Similarly, the minimum cardinality of a set D⊆VD⊆V with |(NG(v)∪{v})∩D|≥k|(NG(v)∪{v})∩D|≥k for all v∈Vv∈V is the kk-tuple domination number γ×k(G)γ×k(G) of GG.Let GG be a graph of order nn and minimum degree δδ and let k∈Nk∈N. We prove that if δ+1ln(δ+1)≥2k, then γk(G)≤nδ+1(kln(δ+1)+∑i=0k−11i!(δ+1)k−1−i) and γ×k(G)≤nδ+1(kln(δ+1)+∑i=0k−1(k−i)i!(δ+1)k−1−i). Furthermore, we prove that if δ≥2δ≥2, then γ×3(G)≤nδ−1(ln(δ−1)+ln(∑v∈VdG(v)+12)−ln(n)+1) which generalizes a recent result of J. Harant and M. Henning.