Domination in graphs of minimum degree at least two and large girth

We prove that for graphs of order n, minimum degree delta >= 2 and girth g >= 5 the graphs of order n and girth g >= 5 the domination number. satisfies gamma <= (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.