Improved bounds for zero-sum cycles in $\mathbb{Z}_p^d$
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
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2024
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| _version_ | 1866917718093987840 |
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| author | Christoph, Micha Knierim, Charlotte Martinsson, Anders Steiner, Raphael |
| author_facet | Christoph, Micha Knierim, Charlotte Martinsson, Anders Steiner, Raphael |
| contents | For a finite Abelian group $(Γ,+)$, let $n(Γ)$ denote the smallest positive integer $n$ such that for each labelling of the arcs of the complete digraph of order $n$ using elements from $Γ$, there exists a directed cycle such that the total sum of the arc-labels along the cycle equals $0$. Alon and Krivelevich initiated the study of the parameter $n(\cdot)$ on cyclic groups and proved that $n(\mathbb{Z}_q)=O(q\log q)$. Studying the prototypical case when $Γ=\mathbb{Z}_p^d$ is a power of a cyclic group of prime order, Letzter and Morrison recently showed that $n(\mathbb{Z}_p^d) \le O(pd(\log d)^2)$ and that $n(\mathbb{Z}_2^d)\le O(d \log d)$. They then posed the problem of proving an (asymptotically optimal) upper bound of $n(\mathbb{Z}_p^d)\le O(pd)$ for all primes $p$ and $d \in \mathbb{N}$. In this paper, we solve this problem for $p=2$ and improve their bound for all primes $p \ge 3$ by proving $n(\mathbb{Z}_2^d)\le 5d$ and $n(\mathbb{Z}_p^d)\le O(pd\log d)$. While the first bound determines $n(\mathbb{Z}_2^d)$ up to a multiplicative error of $5$, the second bound is tight up to a $\log d$ factor. Moreover, our result shows that a tight bound of $n(\mathbb{Z}_p^d)=Θ(pd)$ for arbitrary $p$ and $d$ would follow from a (strong form) of the well-known conjecture of Jaeger, Linial, Payan and Tarsi on additive bases in $\mathbb{Z}_p^d$.
Along the way to proving these results, we establish a generalization of a hypergraph matching result by Haxell in a matroidal setting. Concretely, we obtain sufficient conditions for the existence of matchings in a hypergraph whose hyperedges are labelled by the elements of a matroid, with the property that the edges in the matching induce a basis of the matroid. We believe that these statements are of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_07644 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improved bounds for zero-sum cycles in $\mathbb{Z}_p^d$ Christoph, Micha Knierim, Charlotte Martinsson, Anders Steiner, Raphael Combinatorics For a finite Abelian group $(Γ,+)$, let $n(Γ)$ denote the smallest positive integer $n$ such that for each labelling of the arcs of the complete digraph of order $n$ using elements from $Γ$, there exists a directed cycle such that the total sum of the arc-labels along the cycle equals $0$. Alon and Krivelevich initiated the study of the parameter $n(\cdot)$ on cyclic groups and proved that $n(\mathbb{Z}_q)=O(q\log q)$. Studying the prototypical case when $Γ=\mathbb{Z}_p^d$ is a power of a cyclic group of prime order, Letzter and Morrison recently showed that $n(\mathbb{Z}_p^d) \le O(pd(\log d)^2)$ and that $n(\mathbb{Z}_2^d)\le O(d \log d)$. They then posed the problem of proving an (asymptotically optimal) upper bound of $n(\mathbb{Z}_p^d)\le O(pd)$ for all primes $p$ and $d \in \mathbb{N}$. In this paper, we solve this problem for $p=2$ and improve their bound for all primes $p \ge 3$ by proving $n(\mathbb{Z}_2^d)\le 5d$ and $n(\mathbb{Z}_p^d)\le O(pd\log d)$. While the first bound determines $n(\mathbb{Z}_2^d)$ up to a multiplicative error of $5$, the second bound is tight up to a $\log d$ factor. Moreover, our result shows that a tight bound of $n(\mathbb{Z}_p^d)=Θ(pd)$ for arbitrary $p$ and $d$ would follow from a (strong form) of the well-known conjecture of Jaeger, Linial, Payan and Tarsi on additive bases in $\mathbb{Z}_p^d$. Along the way to proving these results, we establish a generalization of a hypergraph matching result by Haxell in a matroidal setting. Concretely, we obtain sufficient conditions for the existence of matchings in a hypergraph whose hyperedges are labelled by the elements of a matroid, with the property that the edges in the matching induce a basis of the matroid. We believe that these statements are of independent interest. |
| title | Improved bounds for zero-sum cycles in $\mathbb{Z}_p^d$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2407.07644 |