Optimal bounds for zero-sum cycles. I. Odd order

TL;DR

This paper establishes the optimal upper bound for the parameter $n( ext{group})$ in finite groups of odd order, showing it equals the group order plus one, extending previous bounds and fully solving the problem for odd groups.

Contribution

It proves that for all finite groups of odd order, the parameter $n( ext{group})$ equals the group order plus one, confirming the conjectured bound and completing the classification for odd groups.

Findings

Proved $n( ext{group})=| ext{group}|+1$ for all odd order groups.

Confirmed the bound is tight for cyclic groups.

Extended the understanding of zero-sum cycle bounds in group-labeled digraphs.

Abstract

For a finite (not necessarily Abelian) group $(\Gamma,\cdot)$, let $n(\Gamma) \in \mathbb{N}$ denote the smallest positive integer $n$ such that for every labelling of the arcs of the complete digraph of order $n$ using elements from $\Gamma$, there exists a directed cycle such that the arc-labels along the cycle multiply to the identity. Alon and Krivelevich initiated the study of the parameter $n(\cdot)$ on cyclic groups and proved $n(\mathbb{Z}_q)=O(q \log q)$. This was later improved to a linear bound of $n(\Gamma)\le 8|\Gamma|$ for every finite Abelian group by M\'{e}sz\'{a}ros and the last author, and then further to $n(\Gamma)\le 2|\Gamma|-1$ for every non-trivial finite group independently by Berendsohn, Boyadzhiyska and Kozma as well as by Akrami, Alon, Chaudhury, Garg, Mehlhorn and Mehta. In this series of two papers we conclude this line of research by proving that $n(\Gamma)\le |\Gamma|+1$ for every finite group $(\Gamma,\cdot)$, which is the best possible such bound in terms of the group order and precisely determines the value of $n(\Gamma)$ for all cyclic groups as $n(\mathbb{Z}_q)=q+1$. In the present paper we prove the above result for all groups of odd order. The proof for groups of even order needs to overcome substantial additional obstacles and will be presented in the second part of this series.