Improved Upper Bounds on the Domination Number of Graphs With Minimum Degree at Least Five

An algorithmic upper bound on the domination number γ of graphs in terms of the order n and the minimum degree δ is proved. It is demonstrated that the bound improves best previous bounds for any 5 ≤ δ ≤ 50. In particular, for δ = 5, Xing et al. proved in 2006 that γ ≤ 5n/14 < 0.3572n. This bound is improved to 0.3440n. For δ = 6, Clark et al. in 1998 established γ < 0.3377n, while Biró et al. recently improved it to γ < 0.3340n. Here the bound is further improved to γ < 0.3159n. For δ = 7, the best earlier bound 0.3088n is improved to γ < 0.2927n.