The upper bound on k-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 d̃k−1 + 1 δ − k + 2 n, where d̃m = 1 n ∑n i=1 (di+1 m ) with di the degree of the ith vertex of G.