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

Abstract

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 $5\le \delta \le 50$. In particular, for $\delta=5$, Xing et al.\ proved in 2006 that $\gamma \le 5n/14 < 0.3572 n$. This bound is improved to $0.3440 n$. For $\delta=6$, Clark et al.\ in 1998 established $\gamma <0.3377 n$, while Bir\'o et al. recently improved it to $\gamma <0.3340 n$. Here the bound is further improved to $\gamma < 0.3159 n$. For $\delta=7$, the best earlier bound $0.3 088 n$ is improved to $\gamma < 0.2927 n$.