Surprising identities for the greedy independent set on Cayley trees
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2021
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| Acceso en línea: | |
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| _version_ | 1866914670658453504 |
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| author | Contat, Alice |
| author_facet | Contat, Alice |
| contents | We prove a surprising symmetry between the law of the size $G_n$ of the greedy independent set on a uniform Cayley tree $ \mathcal{T}_n$ of size $n$ and that of its complement. We show that $G_n$ has the same law as the number of vertices at even height in $ \mathcal{T}_n$ rooted at a uniform vertex. This enables us to compute the exact law of the $G_n$. We also give a Markovian construction of the greedy independent set, which highlights the symmetry of $G_n$ and whose proof uses a new Markovian exploration of rooted Cayley trees which is of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_03800 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Surprising identities for the greedy independent set on Cayley trees Contat, Alice Probability Combinatorics We prove a surprising symmetry between the law of the size $G_n$ of the greedy independent set on a uniform Cayley tree $ \mathcal{T}_n$ of size $n$ and that of its complement. We show that $G_n$ has the same law as the number of vertices at even height in $ \mathcal{T}_n$ rooted at a uniform vertex. This enables us to compute the exact law of the $G_n$. We also give a Markovian construction of the greedy independent set, which highlights the symmetry of $G_n$ and whose proof uses a new Markovian exploration of rooted Cayley trees which is of independent interest. |
| title | Surprising identities for the greedy independent set on Cayley trees |
| topic | Probability Combinatorics |
| url | https://arxiv.org/abs/2103.03800 |