Independence Complexes of Stable Kneser Graphs

TL;DR

This paper investigates the topological structure of independence complexes in stable Kneser graphs, revealing their homotopy types for small parameters and related graphs, advancing understanding of their combinatorial topology.

Contribution

It determines the homotopy types of independence complexes for SG_{2,k} and related graphs, providing new insights into their topological properties.

Findings

Independence complex of SG_{2,k} is a wedge of 2-spheres.

Homotopy types of independence complexes of related graphs are characterized.

Results contribute to the topological understanding of stable Kneser graphs.

Abstract

For integers n\geq 1, k\geq 0, the stable Kneser graph SG_{n,k} (also called the Schrijver graph) has as vertex set the stable n-subsets of [2n+k] and as edges disjoint pairs of n-subsets, where a stable n-subset is one that does not contain any 2-subset of the form {i,i+1} or {1,2n+k}. The stable Kneser graphs have been an interesting object of study since the late 1970's when A. Schrijver determined that they are a vertex critical class of graphs with chromatic number k+2. This article contains a study of the independence complexes of SG_{n,k} for small values of n and k. Our contributions are two-fold: first, we find that the homotopy type of the independence complex of SG_{2,k} is a wedge of spheres of dimension two. Second, we determine the homotopy types of the independence complexes of certain graphs related to SG_{n,2}.