The upper bound on -tuple domination numbers of graphs

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