On the chromatic number of a subgraph of the Kneser graph

Abstract

Let n and k be positive integers with $n\ge 2k$. Consider a circle C with n points $1,\dots ,n$ in clockwise order. The interlacing graph ${\mathrm{IG}}_{n,k}$ is the graph with vertices corresponding to k-subsets of [n] that do not contain two adjacent points on C, and edges between k-subsets P and Q if they interlace: after removing the points in P from C, the points in Q are in different connected components. In this paper we prove that the circular chromatic number of ${\mathrm{IG}}_{n,k}$ is equal to $n/k$, hence the chromatic number is $\lceil n/k\rceil$, and that its independence number is $\left(\begin{array}{c}n-k-1\\ k-1\end{array}\right)$.