On the domination number of graphs with minimum degree at least seven

Abstract

A dominating set in a graph G is a set $S\subseteq V\left(G\right)$ such that every vertex that is not in S is adjacent to a vertex from it. The domination number $\gamma \left(G\right)$ is the minimum cardinality of a dominating set in G. For $k\ge 1$, let $G_k$ denote the class of all connected graphs with minimum degree at least k. The problem to determine or estimate the best possible constants $c_{\mathrm{dom},k}$ (which depend only on k) such that $\gamma \left(G\right)\le c_{\mathrm{dom},k}\cdot n\left(G\right)$ for all $G\in G_k$ is well studied, where $n\left(G\right)$ denotes the order of G. In this paper we establish best known lower and upper bounds to date on the constant $c_{\mathrm{dom},7}$, namely $\frac{3}{14}\le c_{\mathrm{dom},7}\le \frac{4}{14}$.