Star clusters in independence complexes of graphs

TL;DR

This paper introduces star clusters in simplicial complexes as a new tool to analyze the topology of independence complexes of graphs, providing insights into their homotopy types and connectivity.

Contribution

It develops the concept of star clusters to study independence complexes, addressing open questions and offering a homotopical approach to graph coloring and connectivity.

Findings

Answered questions on homotopy types of independence complexes of triangle-free graphs.

Provided new results on the connectivity of independence complexes.

Presented an alternative method to analyze graph chromatic number using topology.

Abstract

We introduce the notion of \textit{star cluster} of a simplex in a simplicial complex. This concept provides a general tool to study the topology of independence complexes of graphs. We use star clusters to answer a question arisen from works of Engstr\"om and Jonsson on the homotopy type of independence complexes of triangle-free graphs and to investigate a large number of examples which appear in the literature. We present an alternative way to study the chromatic number of a graph from a homotopical point of view and obtain new results regarding the connectivity of independence complexes.