Improved bounds for zero-sum cycles in Zpd

Abstract

For a finite abelian group $(\Gamma ,+)$, let $n\left(\Gamma \right)$ denote the smallest positive integer n such that for each labeling 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 $n\left(Z_q\right)=O(q\log q)$. Several improvements and generalizations of this bound have since been obtained, and an optimal bound in terms of the group order of the form $n\left(\Gamma \right)\le \left|\Gamma \right|+1$ was recently announced by Campbell, Gollin, Hendrey and the last author. While this bound is tight when the group Γ is cyclic, in cases when Γ is far from being cyclic, significant improvements on the bound can be made. In this direction, studying the prototypical case when $\Gamma =Z_p^d$ is a power of a cyclic group of prime order, Letzter and Morrison [Journal of Combinatorial Theory Series B, 2024] showed that $n\left(Z_p^d\right)\le O\left(pd{(\log d)}^2\right)$ and that $n\left(Z_2^d\right)\le O(d\log d)$. They then posed the problem of proving an (asymptotically optimal) upper bound of $n\left(Z_p^d\right)\le O\left(pd\right)$ for all primes p and $d\in N$. In this paper, we solve this problem for $p=2$ and improve their bound for all primes $p\ge 3$ by proving $n\left(Z_2^d\right)\le 5d$ and $n\left(Z_p^d\right)\le O(pd\log d)$. While the first bound determines $n\left(Z_2^d\right)$ 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 $n\left(Z_p^d\right)=\Theta \left(pd\right)$ 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 $Z_p^d$.

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 labeled 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.