[Paper Review] On the circular chromatic number of a subgraph of the Kneser graph

Bart Litjens, Sven Polak+2 more|arXiv (Cornell University)|Mar 12, 2018

Topological and Geometric Data Analysis12 references3 citations

TL;DR

This paper establishes that the circular chromatic number of the r-stable interlacing graph $\mathrm{IG}_{n,k}^{(r)}$ is exactly $n/k$, implying its chromatic number is $\lceil n/k \rceil$. The authors prove this via a graph homomorphism to the circular clique $K_{n/k}$, while also determining the independence number as $\binom{n-(r-1)k-1}{k-1}$ and showing the circular clique number is also $n/k$, strengthening prior results on Kneser-type graphs.

ABSTRACT

Let $n,k,r$ be positive integers with $n \geq rk$ and $r \geq 2$. Consider a circle $C$ with~$n$ points~$1,\ldots,n$ in clockwise order. The $r$-stable \emph{interlacing graph} $ ext{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 \emph{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 $ ext{IG}_{n,k}^{(r)}$ is equal to $ n/k $ (hence the chromatic number is $\lceil n/k ceil$) 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.

Motivation & Objective

• To determine the circular chromatic number of the r-stable interlacing graph $\mathrm{IG}_{n,k}^{(r)}$, a subgraph of the Kneser graph with geometric and combinatorial constraints.
• To establish the chromatic number of $\mathrm{IG}_{n,k}^{(r)}$ as $\lceil n/k \rceil$, extending known results on Kneser and Schrijver graphs.
• To compute the independence number of $\mathrm{IG}_{n,k}^{(r)}$, refining the Erd\'s-Ko-Rado bound for this structured subgraph.
• To show that the circular clique number of $\mathrm{IG}_{n,k}^{(r)}$ is also $n/k$, demonstrating tightness of the circular chromatic number bound.

Proposed method

• Construct a graph homomorphism from $\mathrm{IG}_{n,k}^{(r)}$ to the circular clique $K_{n/k}$ using a vertex labeling $P^j \mapsto \overline{jk}$, relying on cyclic group structure.
• Define $r$-stable $k$-polygons as $k$-subsets of $[n]$ with at least $r-1$ elements between any two points on a circle.
• Use interlacing condition: two $k$-polygons are adjacent if removing one splits the circle into arcs, each containing exactly one point of the other.
• Apply properties of fractional parts $\{x\} = x - \lfloor x \rfloor$ to analyze position relations between polygons and verify adjacency in the homomorphism.
• Prove that non-interlacing pairs correspond exactly to vertices in $K_{n/k}$ within distance less than $k$, ensuring the homomorphism preserves edges.
• Use combinatorial counting and interval analysis to verify that all required interlacing pairs are correctly mapped, especially for $s=1$ and maximal $s$ in the interval $[1, \lfloor(i+1)n/k\rfloor - \lceil in/k\rceil - 1]$.

Experimental results

Research questions

• RQ1 What is the circular chromatic number of the r-stable interlacing graph $\mathrm{IG}_{n,k}^{(r)}$?
• RQ2 How does the independence number of $\mathrm{IG}_{n,k}^{(r)}$ compare to that of the Kneser graph $\mathrm{KG}_{n,k}$, and can it be precisely computed?
• RQ3 Is the circular clique number of $\mathrm{IG}_{n,k}^{(r)}$ equal to its circular chromatic number, indicating tightness of the bound?
• RQ4 Can a graph homomorphism from $\mathrm{IG}_{n,k}^{(r)}$ to $K_{n/k}$ be explicitly constructed using cyclic symmetry and polygonal positioning?

Key findings

• The circular chromatic number of $\mathrm{IG}_{n,k}^{(r)}$ is exactly $n/k$, which implies its chromatic number is $\lceil n/k \rceil$.
• The independence number of $\mathrm{IG}_{n,k}^{(r)}$ is $\binom{n-(r-1)k-1}{k-1}$, providing a tight upper bound for independent sets in this r-stable setting.
• The circular clique number of $\mathrm{IG}_{n,k}^{(r)}$ is also $n/k$, showing that $K_{n/k}$ is a minimal circular clique mapping into the graph.
• The graph homomorphism from $\mathrm{IG}_{n,k}^{(r)}$ to $K_{n/k}$ is explicitly constructed via a cyclic labeling $P^j \mapsto \overline{jk}$, and is proven to preserve adjacency using fractional part analysis.
• The proof establishes that only non-interlacing pairs map to vertices within distance less than $k$ in $K_{n/k}$, confirming the homomorphism is well-defined.
• The construction holds under the condition $\gcd(n,k) = 1$, and the result extends to general $n,k$ via reduction to the coprime case.