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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 $2k+3$.

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Let X ϭ ͕ 1 , 2 , . . . , n ͖ be a set of n elements and let X ( r ) be the collection of all the subsets of X containing precisely r elements . Then the generalised Kneser graph K ( n , r , s ) (when 2 r Ϫ s р n ) is the graph with vertex set X ( r ) and edges AB for A , B X ( r ) with ͉ A ʝ B ͉ р s . Here we show that the odd girth of the generalised Kneser graph K ( n , r , s ) isprovided that n is large enough compared with r and s .

In this paper, we give a generalization of a well-known result of Dirac that given any k vertices in a k-connected graph where k 2, there is a circuit containing all of them. We also generalize a result of Ha ggkvist and Thomassen. Our main result partially answers an open matroid question of Oxley. Academic Press