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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2407.17363 |
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| _version_ | 1866910540352192512 |
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| author | Kelly, Tom Postle, Luke |
| author_facet | Kelly, Tom Postle, Luke |
| contents | We prove that if $G$ is a graph and $f(v) \leq 1/(d(v) + 1/2)$ for each $v\in V(G)$, then either $G$ has an independent set of size at least $\sum_{v\in V(G)}f(v)$ or $G$ contains a clique $K$ such that $\sum_{v\in K}f(v) > 1$. This result implies that for any $σ\leq 1/2$, if $G$ is a graph and every clique $K\subseteq V(G)$ has at most $(1 - σ)(|K| - σ)$ simplicial vertices, then $α(G) \geq \sum_{v\in V(G)} 1 / (d(v) + 1 - σ)$. Letting $σ= 0$ implies the famous Caro-Wei Theorem, and letting $σ= 1/2$ implies that if fewer than half of the vertices in each clique of $G$ are simplicial, then $α(G) \geq \sum_{v\in V(G)}1/(d(v) + 1/2)$, which is tight for the 5-cycle. When applied to the complement of a graph, this result implies the following new Tur\' an stability result. If $G$ is a $K_{r + 1}$-free graph with more than $(1 - 1/r)n^2/2 - n/4$ edges, then $G$ contains an independent set $I$ such that at least half of the vertices in $I$ are complete to $G - I$. Applying this stability result iteratively provides a new proof of the stability version of Tur\' an's Theorem in which $K_{r + 1}$-free graphs with close to the extremal number of edges are $r$-partite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_17363 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improving the Caro-Wei bound and applications to Turán stability Kelly, Tom Postle, Luke Combinatorics We prove that if $G$ is a graph and $f(v) \leq 1/(d(v) + 1/2)$ for each $v\in V(G)$, then either $G$ has an independent set of size at least $\sum_{v\in V(G)}f(v)$ or $G$ contains a clique $K$ such that $\sum_{v\in K}f(v) > 1$. This result implies that for any $σ\leq 1/2$, if $G$ is a graph and every clique $K\subseteq V(G)$ has at most $(1 - σ)(|K| - σ)$ simplicial vertices, then $α(G) \geq \sum_{v\in V(G)} 1 / (d(v) + 1 - σ)$. Letting $σ= 0$ implies the famous Caro-Wei Theorem, and letting $σ= 1/2$ implies that if fewer than half of the vertices in each clique of $G$ are simplicial, then $α(G) \geq \sum_{v\in V(G)}1/(d(v) + 1/2)$, which is tight for the 5-cycle. When applied to the complement of a graph, this result implies the following new Tur\' an stability result. If $G$ is a $K_{r + 1}$-free graph with more than $(1 - 1/r)n^2/2 - n/4$ edges, then $G$ contains an independent set $I$ such that at least half of the vertices in $I$ are complete to $G - I$. Applying this stability result iteratively provides a new proof of the stability version of Tur\' an's Theorem in which $K_{r + 1}$-free graphs with close to the extremal number of edges are $r$-partite. |
| title | Improving the Caro-Wei bound and applications to Turán stability |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2407.17363 |