Star clusters in independence complexes of graphs

Details Star clusters in independence complexes of graphs Star clusters in independence complexes of graphs

2010-07-02

arxiv.org/abs/1007.0418v1

Mathematics

Combinatorics

22 pages, 13 figures

Scientific paper

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.

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