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arXiv · 1912.12605

Complexes of graphs with bounded independence number

Minki Kim·Alan Lew
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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 $n$.

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Minki Kim, Alan Lew. 2019-12-29. Complexes of graphs with bounded independence number. https://doi.org/10.1007/s11856-022-2308-4

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