Improved upper bounds for the k-tuple domination number

We improve the generalized upper bound for the k-tuple domination number given in [A. Gagarin and V.E. Zverovich, A generalized upper bound for the k-tuple domination number, Discrete Math. 308 no. 5–6 (2008), 880–885]. Precisely, we show that for any graph G, when k = 3, or k = 4 and d ≤ 3.2, γ×k(G) ≤ ln(δ−k + 2) + ln ( (k − 2)d + ∑k−2 m=2 (k−m) 4min{m, k−2−m} d̂m + d̂k−1 ) + 1 δ − k + 2 n, and, when k = 4 and d > 3.2, or k ≥ 5, γ×k(G) ≤ ln(δ − k + 2) + ln (∑k−2 m=0 (k−m) 4min{m, k−2−m} d̂m + d̂k−1 ) + 1 δ − k + 2 n, where γ×k(G) is the k-tuple domination number, δ is the minimum degree, d is the average degree, and d̂m is the m-degree of G. Moreover, when k ≥ 5, the latter bound can be improved to γ×k(G) ≤ ln(δ − k + 2) + ln (∑k−2 m=0 (k−m) P(k−2,m) d̂m + d̂k−1 ) + 1 δ − k + 2 n, where the coefficient P(t,m) = t t mm(t−m)t−m for t > m > 0, P(t, 0) = P(t, t) = 1, with t = k − 2.