Independent dominating sets in graphs of girth five

Let γ(G) and γ◦(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 γ(G) ≤ n d (log d+ 1) In this paper, the main result is that if G is any n-vertex d-regular graph of girth at least five then γ◦(G) ≤ n d (log d+ c) for some constant c independent of d. This result is sharp in the sense that as d → ∞, almost all d-regular n-vertex graphs G of girth at least five have γ◦(G) ∼ n d log d. Furthermore, if G is a disjoint union of n 2d complete bipartite graphs Kd,d, then γ◦(G) = 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 γ◦(G) ∼ n2 as d→∞. Therefore both the girth and regularity conditions are required for the main result.