Zero sum cycles in complete digraphs

TL;DR

This paper investigates zero-sum cycles in complete directed graphs labeled with elements of finite Abelian groups, improving bounds on the minimal graph size needed for guaranteed zero-sum cycles, and applies these results to graph minors.

Contribution

It establishes linear bounds on the minimal size of complete digraphs needed for zero-sum cycles in finite Abelian groups, improving previous logarithmic bounds.

Findings

Proves that n(Z_q) is linearly bounded by q.

Shows n(A) ≤ 8|A| for any finite Abelian group A.

Demonstrates every K_{16q}-minor contains a cycle divisible by q.

Abstract

Given a non-trivial finite Abelian group $(A,+)$, let $n(A) \ge 2$ be the smallest integer such that for every labelling of the arcs of the bidirected complete graph of order $n(A)$ with elements from $A$ there exists a directed cycle for which the sum of the arc-labels is zero. The problem of determining $n(\mathbb{Z}_q)$ for integers $q \ge 2$ was recently considered by Alon and Krivelevich, who proved that $n(\mathbb{Z}_q)=O(q \log q)$. Here we improve their bound and show that $n(\mathbb{Z}_q)$ grows linearly. More generally we prove that for every finite Abelian group $A$ we have $n(A) \le 8|A|$, while if $|A|$ is prime then $n(A) \le \frac{3}{2}|A|$. As a corollary we also obtain that every $K_{16q}$-minor contains a cycle of length divisible by $q$ for every integer $q \ge 2$, which improves a result by Alon and Krivelevich.