TITLE: Bound on graph domination number when min degree is 7
QUESTION [5 upvotes]: I have a graph $G$ whose minimum vertex degree is $\delta=7$.
I am seeking an upper bound on the domination number $\gamma(G)$
in terms of the number of vertices $n$ of $G$.
I found a paper by
Edwin Clark, Boris Shekhtman, Stephen Suen, and David Fisher, 
"Upper bounds for the domination number of a graph," Congressus 
Numerantium, 132:99–124, 1998 (CiteSeer link),
that implies a bound of about $0.31 n$ (more precisely, $(18286568 / 58640175)n$).
I was hoping for a smaller upper bound.  Perhaps there have been advances since that paper?
Any pointers to relevant literature would be much appreciated.  Thanks!

REPLY [5 votes]: See this article, it provides the best bound till date:
Bujtás, Csilla; Klavžar, Sandi, Improved upper bounds on the domination number of graphs with minimum degree at least five, Graphs Comb. 32, No. 2, 511-519 (2016). ZBL1339.05280.

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 $5 \leq \delta \leq 50$. In particular, for $\delta =5$, Xing et al. (Graphs Comb. 22:127–143, 2006) proved that $\gamma \leq 5n/14 < 0.3572n$. This bound is improved to $0.3440 n$. For $\delta=6$, Clark et al. (Congr. Numer. 132:99–123, 1998) established $\gamma < 0.3377n$, while Biró et al. (Bull. Inst. Comb. Appl. 64:73–83, 2012) recently improved it to $\gamma < 0.3340n$. Here the bound is further improved to $\gamma < 0.3159n$. For $\delta=7$, the best earlier bound $0.3088n$ is improved to $\gamma<0.2927n$.