Minimum degree conditions for graph rigidity
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916532470153216 |
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| author | Krivelevich, Michael Lew, Alan Michaeli, Peleg |
| author_facet | Krivelevich, Michael Lew, Alan Michaeli, Peleg |
| contents | We study minimum degree conditions that guarantee that an $n$-vertex graph is rigid in $\mathbb{R}^d$. For small values of $d$, we obtain a tight bound: for $d = O(\sqrt{n})$, every $n$-vertex graph with minimum degree at least $(n+d)/2 - 1$ is rigid in $\mathbb{R}^d$. For larger values of $d$, we achieve an approximate result: for $d = O(n/{\log^2}{n})$, every $n$-vertex graph with minimum degree at least $(n+2d)/2 - 1$ is rigid in $\mathbb{R}^d$. This bound is tight up to a factor of two in the coefficient of $d$.
As a byproduct of our proof, we also obtain the following result, which may be of independent interest: for $d = O(n/{\log^2}{n})$, every $n$-vertex graph with minimum degree at least $d$ has pseudoachromatic number at least $d+1$; namely, the vertex set of such a graph can be partitioned into $d+1$ subsets such that there is at least one edge between each pair of subsets. This is tight. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_14364 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimum degree conditions for graph rigidity Krivelevich, Michael Lew, Alan Michaeli, Peleg Combinatorics 05C10, 52C25 We study minimum degree conditions that guarantee that an $n$-vertex graph is rigid in $\mathbb{R}^d$. For small values of $d$, we obtain a tight bound: for $d = O(\sqrt{n})$, every $n$-vertex graph with minimum degree at least $(n+d)/2 - 1$ is rigid in $\mathbb{R}^d$. For larger values of $d$, we achieve an approximate result: for $d = O(n/{\log^2}{n})$, every $n$-vertex graph with minimum degree at least $(n+2d)/2 - 1$ is rigid in $\mathbb{R}^d$. This bound is tight up to a factor of two in the coefficient of $d$. As a byproduct of our proof, we also obtain the following result, which may be of independent interest: for $d = O(n/{\log^2}{n})$, every $n$-vertex graph with minimum degree at least $d$ has pseudoachromatic number at least $d+1$; namely, the vertex set of such a graph can be partitioned into $d+1$ subsets such that there is at least one edge between each pair of subsets. This is tight. |
| title | Minimum degree conditions for graph rigidity |
| topic | Combinatorics 05C10, 52C25 |
| url | https://arxiv.org/abs/2412.14364 |