Homotopy type of the independence complex of some categorical products of graphs

TL;DR

This paper proves a conjecture about the homotopy type of independence complexes of certain graph products, showing they are wedges of spheres and computing their types for specific cases.

Contribution

It confirms a conjecture on the homotopy type of independence complexes of categorical graph products and extends calculations to new classes of graphs.

Findings

Homotopy type of $K_2\times K_3\times K_n$ is a wedge of $(n-1)(3n-2)$ spheres of dimension 3.

Homotopy types of $C_{3r}\times K_n$ and $K_2\times K_m\times K_n$ are explicitly calculated.

For $C_m \times K_n$ with $m$ not multiple of 3, the homotopy type is a wedge of spheres of at most 2 dimensions and possibly Moore spaces.

Abstract

It was conjectured by Goyal, Shukla and Singh that the independence complex of the categorical product $K_2\times K_3\times K_n$ has the homotopy type of a wedge of $(n-1)(3n-2)$ spheres of dimension $3$. Here we prove this conjecture by calculating the homotopy type of the independence complex of the graphs $C_{3r}\times K_n$ and $K_2\times K_m\times K_n$. For $C_m \times K_n$ when $m$ is not a multiple of $3$, we calculate the homotopy type for $m = 4, 5$ and show that for other values it has to have the homotopy type of a wedge of spheres of at most $2$ consecutive dimensions and maybe some Moore spaces.