On the bandwidth of the Kneser graph

Abstract

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 bandwidthB(G) of G to be the minimum possible value of dilation(f) over all such one to one maps f. Next define the Kneser GraphK(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={AB:A,B∈[n]r,A∩B=∅}. For fixed r≥4 and n growing we show that B(K(n,r))=n−1r+12n−4r−1−12n−1r−2+O(nr−4).