The equivariant topology of stable Kneser graphs

TL;DR

This paper investigates the topological properties of stable Kneser graphs, revealing their homotopy types under group actions and identifying which graphs serve as homotopy test graphs for chromatic number bounds.

Contribution

It determines the equivariant homotopy type of the Hom complex of stable Kneser graphs and characterizes when these graphs are homotopy test graphs.

Findings

Stable Kneser graphs $SG_{2s,4}$ are homotopy test graphs.

For certain parameters, $SG_{n,k}$ are not homotopy test graphs.

The paper almost fully determines the $(C_2\times D_{2m})$-homotopy type of the Hom complex.

Abstract

The stable Kneser graph $SG_{n,k}$, $n\ge1$, $k\ge0$, introduced by Schrijver \cite{schrijver}, is a vertex critical graph with chromatic number $k+2$, its vertices are certain subsets of a set of cardinality $m=2n+k$. Bj\"orner and de Longueville \cite{anders-mark} have shown that its box complex is homotopy equivalent to a sphere, $\Hom(K_2,SG_{n,k})\homot\Sphere^k$. The dihedral group $D_{2m}$ acts canonically on $SG_{n,k}$, the group $C_2$ with 2 elements acts on $K_2$. We almost determine the $(C_2\times D_{2m})$-homotopy type of $\Hom(K_2,SG_{n,k})$ and use this to prove the following results. The graphs $SG_{2s,4}$ are homotopy test graphs, i.e. for every graph $H$ and $r\ge0$ such that $\Hom(SG_{2s,4},H)$ is $(r-1)$-connected, the chromatic number $\chi(H)$ is at least $r+6$. If $k\notin\set{0,1,2,4,8}$ and $n\ge N(k)$ then $SG_{n,k}$ is not a homotopy test graph, i.e.\ there are a graph $G$ and an $r\ge1$ such that $\Hom(SG_{n,k}, G)$ is $(r-1)$-connected and $\chi(G)<r+k+2$.