The upper bound on kk-tuple domination numbers of graphs

In a graph GG, a vertex is said to dominate itself and all vertices adjacent to it. For a positive integer kk, the kk-tuple domination number γ×k(G)γ×k(G) of GG is the minimum size of a subset DD of V(G)V(G) such that every vertex in GG is dominated by at least kk vertices in DD. To generalize/improve known upper bounds for the kk-tuple domination number, this paper establishes that for any positive integer kk and any graph GG of nn vertices and minimum degree δδ, γ×k(G)≤ln(δ−k+2)+lnd˜k−1+1δ−k+2n, where d˜m=1n∑i=1ndi+1m with didi the degree of the iith vertex of GG.