The Independence Number of Dense Graphs with Large Odd Girth

Abstract

Let $G$ be a graph with $n$ vertices and odd girth $2k+3$. Let the degree of a vertex $v$ of $G$ be $d_1 (v)$. Let $\alpha (G)$ be the independence number of $G$. Then we show $\alpha (G) \geq 2^{-\left({{k-1}\over {k}}\right)} \left[ \displaystyle{\sum_{v\in G}} d_1 (v)^{{{1}\over {k-1}}} \right]^{(k-1)/k}$. This improves and simplifies results proven by Denley.