On the circular chromatic number of a subgraph of the Kneser graph - Litjens, Bart; Polak, Sven; Sevenster, Bart; Vena, Lluís (2018)
Let $n,k,r$ be positive integers with $n ≥ rk$ and $r ≥ 2$. Consider a circle $C$ with~$n$ points~$1,…,n$ in clockwise order. The $r$-stable interlacing graph $\text{IG}_{n,k}^{(r)}$ is the graph with vertices corresponding to $k$-subsets $S$ of $\{1,...,n\}$ such that any two distinct points in~$S$ have distance at least~$r$ around the circle, 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 $\text{IG}_{n,k}^{(r)}$ is equal to $ n/k $ (hence the chromatic number is $\lceil n/k \rceil$) and that its circular clique number is also $ n/k $. Furthermore, we show that its independence number is $\binom{n-(r-1)k-1}{k-1}$, thereby strengthening a result by Talbot.