Homotopy type of Neighborhood Complexes of Kneser graphs, $KG_{2,k}$

TL;DR

This paper determines the homotopy type of the neighborhood complex of the Kneser graph $KG_{2,k}$ as a wedge of spheres, extending known results for stable Kneser graphs and introducing a maximal subgraph with similar properties.

Contribution

It proves the homotopy type of the neighborhood complex of $KG_{2,k}$ as a wedge of spheres and constructs a maximal subgraph whose neighborhood complex deformation retracts onto that of the stable Kneser graph.

Findings

Homotopy type of neighborhood complex of $KG_{2,k}$ is a wedge of $(k+4)(k+1)+1$ spheres of dimension $k$.

Constructed a maximal subgraph $S_{2,k}$ with neighborhood complex homotopy equivalent to that of $SG_{2,k}$.

Neighborhood complex of $S_{2,k}$ deformation retracts onto the neighborhood complex of $SG_{2,k}$.

Abstract

Schrijver identified a family of vertex critical subgraphs of the Kneser graphs called the stable Kneser graphs $SG_{n,k}$. Bj\"{o}rner and de Longueville proved that the neighborhood complex of the stable Kneser graph $SG_{n,k}$ is homotopy equivalent to a $k-$sphere. In this article, we prove that the homotopy type of the neighborhood complex of the Kneser graph $KG_{2,k}$ is a wedge of $(k+4)(k+1)+1$ spheres of dimension $k$. We construct a maximal subgraph $S_{2,k}$ of $KG_{2,k}$, whose neighborhood complex is homotopy equivalent to the neighborhood complex of $SG_{2,k}$. Further, we prove that the neighborhood complex of $S_{2,k}$ deformation retracts onto the neighborhood complex of $SG_{2,k}$.