Homology, Homotopy and Applications

Volume 26 (2024)

Number 2

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

Pages: 349 – 373

DOI: https://dx.doi.org/10.4310/HHA.2024.v26.n2.a17

Authors

Omar Antolín Camarena (Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, Ciudad de México, México)

Andrés Carnero Bravo (Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, Ciudad de México, México)

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.

Keywords

independence complex, homotopy type

2010 Mathematics Subject Classification

05C76, 05E45, 55P10, 55P15

The full text of this article is unavailable through your IP address: 172.17.0.1

Received 17 October 2023

Received revised 30 January 2024

Accepted 30 January 2024

Published 9 October 2024