Directed cycles with zero weight in Zpk

Abstract

For a finite abelian group A, define $f\left(A\right)$ to be the minimum integer such that for every complete digraph Γ on f vertices and every map $w:E\left(\Gamma \right)\to A$, there exists a directed cycle C in Γ such that ${\sum }_{e\in E\left(C\right)}w\left(e\right)=0$. The study of $f\left(A\right)$ was initiated by Alon and Krivelevich (2021). In this article, we prove that $f\left(Z_p^k\right)=O\left(pk{(\log k)}^2\right)$, where p is prime, with an improved bound of $O(k\log k)$ when $p=2$. These bounds are tight up to a factor which is polylogarithmic in k.