Directed cycles with zero weight in ${\mathbb{Z}}_p^k$

Abstract

For a finite abelian group $A$, define $f(A)$ to be the minimum integer such that for every complete digraph $\Gamma$ on $f$ vertices and every map $w:E(\Gamma )\to A$, there exists a directed cycle $C$ in $\Gamma$ such that ${\sum }_{e\in E(C)}w(e)=0$. The study of $f(A)$ was initiated by Alon and Krivelevich (2021). In this article, we prove that $f({\mathbb{Z}}_p^k)=O(pk(\log k{)}^2)$, 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 $k$.

18 pages (including a 3 page appendix)