Improved upper bounds on the domination number of graphs with minimum degree at least five

An algorithmic upper bound on the domination number $\gamma$ of graphs in terms of the order $n$ and the minimum degree $\delta$ is proved. It is demonstrated that the bound improves best previous bounds for any $5\le \delta \le 50$\mathrm{.}Inparticular,for$\delta=5$,Xingetal\mathrm{.}provedin2006that$\gamma \le 5n/14 < 0.3572 n$\mathrm{.}Thisboundisimprovedto$0.3440 n$\mathrm{.}For$\delta=6$,Clarketal\mathrm{.}in1998established$\gamma <0.3377 n$, while Bir'o et al. recently improved it to $\gamma <0.3340n$. Here the bound is further improved to $\gamma <0.3159n$. For $\delta =7$, the best earlier bound $0.3 088 n$isimprovedto$\gamma < 0.2927 n$.