On the Bandwidth of the Kneser Graph

Abstract

Let $G = (V,E)$ be a graph on $n$ vertices and $f: V\rightarrow [1,n]$ a one to one map of $V$ onto the integers $1$ through $n$. Let $dilation(f) =$ max$\{ |f(v) - f(w)|: vw\in E \}$. Define the {\it 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 {\it Kneser Graph} $K(n,r)$ to be the graph with vertex set $\binom{[n]}{r}$, the collection of $r$-subsets of an $n$ element set, and edge set $E = \{ vw: v,w\in \binom{[n]}{r}, v\cap w = \emptyset \}$. For fixed $r\geq 4$ and $n\rightarrow \infty$ we show that $$B(K(n,r)) = \binom{n}{r} - \frac{1}{2}\binom{n-1}{r-1} - 2\frac{n^{r-2}}{(r-2)!} + (r + 2)\frac{n^{r-3}}{(r-3)!} + O(n^{r-4}).$$