The Independence Number of Graphs with Large Odd Girth

Let $G$ be an $r$-regular graph of order $n$ and independence number $\alpha(G)$. We show that if $G$ has odd girth $2k+3$ then $\alpha(G)\geq n^{1-1/k}r^{1/k}$. We also prove similar results for graphs which are not regular. Using these results we improve on the lower bound of Monien and Speckenmeyer, for the independence number of a graph of order $n$ and odd girth $2k+3$.