The k-tuple domination number revisited

The following fundamental result for the domination number γ(G) of a graph G was proved by Alon and Spencer, Arnautov, Lovász and Payan: γ(G) ≤ ln(δ + 1) + 1 δ + 1 n, where n is the order and δ is the minimum degree of vertices of G. A similar upper bound for the double domination number was found by Harant and Henning [On double domination in graphs. Discuss. Math. Graph Theory 25 (2005) 29–34], and for the triple domination number by Rautenbach and Volkmann [New bounds on the k-domination number and the k-tuple domination number. Applied Math. Letters 20 (2007) 98–102], who also posed the interesting conjecture on the k-tuple domination number: for any graph G with δ ≥ k − 1, γ×k(G) ≤ ln(δ − k + 2) + ln(d̂k−1 + d̂k−2) + 1 δ − k + 2 n, where d̂m = ∑n i=1 ( di m ) /n is the m-degree of G. This conjecture, if true, would generalise all the mentioned upper bounds and improve an upper bound proved in [A. Gagarin and V. Zverovich, A generalised upper bound for the k-tuple domination number. Discrete Math. (to appear)]. In this paper, we prove Rautenbach–Volkmann’s conjecture.