The odd girth of the generalised Kneser graph

Let X = {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 2r - s less than or equal to n) is the graph with vertex set X-(r) and edges AB for A, B is an element of X-(r) with \A boolean AND B\ less than or equal to s. Here we show that the odd girth of the generalised Kneser graph K(n, r, s) is 2 inverted right perpendicular r-s/n-2(r-s) inverted left perpendicular +1 provided that n is large enough compared with r and s. (C) 1997 Academic Press Limited.