Complexes of graphs with bounded independence number
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2019
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| _version_ | 1866913542137970688 |
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| author | Kim, Minki Lew, Alan |
| author_facet | Kim, Minki Lew, Alan |
| contents | 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 $Δ$. As an application, we obtain the following result:
Let $G$ be a claw-free graph with maximum degree at most $Δ$. Then, every collection of $\left\lfloor\left(\fracΔ{2}+1\right)(n-1)\right\rfloor+1$ independent sets in $G$ has a rainbow independent set of size $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1912_12605 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Complexes of graphs with bounded independence number Kim, Minki Lew, Alan Combinatorics 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 $Δ$. As an application, we obtain the following result: Let $G$ be a claw-free graph with maximum degree at most $Δ$. Then, every collection of $\left\lfloor\left(\fracΔ{2}+1\right)(n-1)\right\rfloor+1$ independent sets in $G$ has a rainbow independent set of size $n$. |
| title | Complexes of graphs with bounded independence number |
| topic | Combinatorics |
| url | https://arxiv.org/abs/1912.12605 |