On the bandwidth of the Kneser graph

Let G = (V,E) be a graph on n vertices and f : V → [1, n] a one to one map of V onto the integers 1 through n. Let dilation(f) = max{|f(v)− f(w)| : vw ∈ E}. Define the bandwidth B(G) of G to be the minimum possible value of dilation(f) over all such one to one maps f . Next define the Kneser Graph K(n, r) to be the graph with vertex set ( [n] r ) , the collection of r-subsets of an n element set, and edge set E = {vw : v, w ∈ ( [n] r ) , v ∩ w = ∅}. For fixed r ≥ 4 and n→∞ we show that