Domination in Graphs of Minimum Degree at least Two and Large Girth

We prove that for graphs of order n, minimum degree δ ≥ 2 and girth g ≥ 5 the domination number γ satisfies γ ≤ ( 1 3 + 2 3g ) n. As a corollary this implies that for cubic graphs of order n and girth g ≥ 5 the domination number γ satisfies γ ≤ ( 44 135 + 82 135g ) n which improves recent results due to Kostochka and Stodolsky (An upper bound on the domination number of n-vertex connected cubic graphs, manuscript (2005)) and Kawarabayashi, Plummer and Saito (Domination in a graph with a 2-factor, J. Graph Theory 52 (2006), 1-6) for large enough girth. Furthermore, it confirms a conjecture due to Reed about connected cubic graphs (Paths, stars and the number three, Combin. Prob. Comput. 5 (1996), 267-276) for girth at least 83.