New bounds on a generalization of Tuza's conjecture
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916657505501184 |
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| author | Parker, Alex |
| author_facet | Parker, Alex |
| contents | For a $k$-uniform hypergraph $H$, let $ν^{(m)}(H)$ denote the maximum size of a set $S$ of edges of $H$ whose pairwise intersection has size less than $m$. Let $τ^{(m)}(H)$ denote the minimum size of a set $S$ of $m$-sets of $V(H)$ such that every edge of $H$ contains some $m$-set from $S$. A conjecture by Aharoni and Zerbib, which generalizes a conjecture of Tuza on the size of minimum edge covers of triangles of a graph, states that for a $k$-uniform hypergraph $H$, $τ^{(k - 1)}(H)/ν^{(k - 1)}(H) \leq \left \lceil \frac{k + 1}{2} \right \rceil$. In this paper, we show that this generalization of Tuza's conjecture holds when $ν^{(k - 1)}(H) \leq 3$. As a corollary, we obtain a graph class which satisfies Tuza's conjecture. We also prove various bounds on $τ^{(m)}(H)/ν^{(m)}(H)$ for other values of $m$ as well as some bounds on the fractional analogues of these numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_06501 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New bounds on a generalization of Tuza's conjecture Parker, Alex Combinatorics 05C65, 05D15 For a $k$-uniform hypergraph $H$, let $ν^{(m)}(H)$ denote the maximum size of a set $S$ of edges of $H$ whose pairwise intersection has size less than $m$. Let $τ^{(m)}(H)$ denote the minimum size of a set $S$ of $m$-sets of $V(H)$ such that every edge of $H$ contains some $m$-set from $S$. A conjecture by Aharoni and Zerbib, which generalizes a conjecture of Tuza on the size of minimum edge covers of triangles of a graph, states that for a $k$-uniform hypergraph $H$, $τ^{(k - 1)}(H)/ν^{(k - 1)}(H) \leq \left \lceil \frac{k + 1}{2} \right \rceil$. In this paper, we show that this generalization of Tuza's conjecture holds when $ν^{(k - 1)}(H) \leq 3$. As a corollary, we obtain a graph class which satisfies Tuza's conjecture. We also prove various bounds on $τ^{(m)}(H)/ν^{(m)}(H)$ for other values of $m$ as well as some bounds on the fractional analogues of these numbers. |
| title | New bounds on a generalization of Tuza's conjecture |
| topic | Combinatorics 05C65, 05D15 |
| url | https://arxiv.org/abs/2406.06501 |