We notice that Haynes-Hedetniemi-Slater Conjecture is true (i.e. $\gamma(G) \leq \frac{ \delta}{ 3\delta -1}n$ for every graph $G$ of size $n$ with minimum degree $\delta \geq 4$, where $\gamma(G)$ is the domination number of $G$). Because the conjecture for $\delta =6$ follows from the estimate n (1 - \prod_{ i= 1}^[\delta + 1} (\delta i)/(\delta i + 1) by W. E. Clark, B. Shekhtman, S. Suen [Upper bounds of the Domination Number of a Graph, Congressus Numerantium, 132 (1998), pp. 99-123.]
Categories: math.GR