Given a non-trivial finite Abelian group $(A,+)$, let $n\left(A\right)\ge 2$ be the smallest integer such that for every labelling of the arcs of the bidirected complete graph ${\overleftrightarrow{K}}_{n\left(A\right)}$ with elements from $A$ there exists a directed cycle for which the sum of the arc-labels is zero. The problem of determining $n\left({\mathbb{Z}}_q\right)$ for integers $q\ge 2$ was recently considered by Alon and Krivelevich, (2020), who proved that $n\left({\mathbb{Z}}_q\right)=O(qlogq)$. Here we improve their result and show that $n\left({\mathbb{Z}}_q\right)$ grows linearly. More generally we prove that for every finite Abelian group $A$ we have $n\left(A\right)\le 8\left|A\right|$, while if $\left|A\right|$ is prime then $n\left(A\right)\le \frac{3}{2}\left|A\right|$.

As a corollary we obtain that every $K_{16q}$-minor contains a cycle of length divisible by $q$ for every integer $q\ge 2$, which improves a result from Alon and Krivelevich, (2020).

给定一个非平凡的有限阿贝尔群 $(\mathrm{一种},+)$， 让 $n\left(\mathrm{一种}\right)\ge 2$ 是最小整数，使得对于双向完全图的弧的每个标记 ${\overleftrightarrow{钾}}_{n\left(\mathrm{一种}\right)}$ 与元素来自 $\mathrm{一种}$存在一个有向循环，其中弧标签的总和为零。确定的问题$n\left({\mathbb{Z}}_q\right)$ 对于整数 $q\ge 2$ 最近被 Alon 和 Krivelevich (2020) 考虑，他们证明了 $n\left({\mathbb{Z}}_q\right)=哦(q日志q)$. 在这里，我们改进了他们的结果并表明$n\left({\mathbb{Z}}_q\right)$线性增长。更一般地，我们证明对于每个有限阿贝尔群$\mathrm{一种}$ 我们有 $n\left(\mathrm{一种}\right)\le 8\left|\mathrm{一种}\right|$, 而如果 $\left|\mathrm{一种}\right|$ 那么是素数 $n\left(\mathrm{一种}\right)\le \frac{3}{2}\left|\mathrm{一种}\right|$.

作为推论，我们得到每个 ${钾}_{16q}$-minor 包含一个长度可被整除的循环 $q$ 对于每个整数 $q\ge 2$，这改进了 Alon 和 Krivelevich (2020) 的结果。