Independent dominating sets in graphs of girth five

Let $\gamma(G)$ and $${\gamma _ \circ }(G)$$ denote the sizes of a smallest dominating set and smallest independent dominating set in a graph G, respectively. One of the first results in probabilistic combinatorics is that if G is an n-vertex graph of minimum degree at least d, then$$\begin{equation}\gamma(G) \leq \frac{n}{d}(\log d + 1).\end{equation}$$In this paper the main result is that if G is any n-vertex d-regular graph of girth at least five, then$$\begin{equation}\gamma_(G) \leq \frac{n}{d}(\log d + c)\end{equation}$$for some constant c independent of d. This result is sharp in the sense that as $d \rightarrow \infty$ , almost all d-regular n-vertex graphs G of girth at least five have$$\begin{equation}\gamma_(G) \sim \frac{n}{d}\log d.\end{equation}$$Furthermore, if G is a disjoint union of ${n}/{(2d)}$ complete bipartite graphs $K_{d,d}$ , then ${\gamma_\circ}(G) = \frac{n}{2}$ . We also prove that there are n-vertex graphs G of minimum degree d and whose maximum degree grows not much faster than d log d such that ${\gamma_\circ}(G) \sim {n}/{2}$ as $d \rightarrow \infty$ . Therefore both the girth and regularity conditions are required for the main result.