Complexes of graphs with bounded independence number

TL;DR

This paper investigates the collapsibility of complexes formed from graphs with bounded independence number, providing new bounds and properties for such complexes, especially in claw-free graphs with bounded degree.

Contribution

It introduces bounds on collapsibility numbers of complexes associated with graphs of bounded maximum degree and derives a rainbow independent set result for claw-free graphs.

Findings

Bounds on collapsibility numbers for complexes of graphs with bounded degree

A new rainbow independent set theorem for claw-free graphs

Characterization of simplicial complexes related to graph independence

Abstract

Let $G=(V,E)$ be a graph and $n$ a positive integer. Let $I_n(G)$ be the abstract simplicial complex whose simplices are the subsets of $V$ that do not contain an independent set of size $n$ in $G$. We study the collapsibility numbers of the complexes $I_n(G)$ for various classes of graphs, focusing on the class of graphs with maximum degree bounded by $\Delta$. As an application, we obtain the following result: Let $G$ be a claw-free graph with maximum degree at most $\Delta$. Then, every collection of $\left\lfloor\left(\frac{\Delta}{2}+1\right)(n-1)\right\rfloor+1$ independent sets in $G$ has a rainbow independent set of size $n$.