A generalised upper bound for the k-tuple domination number

In this paper, we provide an upper bound for the k-tuple domination number that generalises known upper bounds for the double and triple domination numbers. We prove that for any graph G,γ×k(G)⩽ln(δ-k+2)+ln(∑m=1k-1(k-m)d^m+ε)+1δ-k+2n,where γ×k(G)γ×k(G) is the k  -tuple domination number; δδ is the minimal degree; d^m is the m-degree of G  ; ε=1ε=1 if k=1k=1 or 2 and ε=-dε=-d if k⩾3k⩾3; d is the average degree.