AI 中文总结
本文研究正则有向图中的零和环，证明了零和环存在的阈值，给出了顶点不交与边不交零和环的数量下界，结果渐近最优，相关结论可推广至欧拉有向图，所用技术基于Friedland的行列式-永久式论证。
AI 中文摘要
设Γ是阶数k≥2的有限群，将简单无环d-正则有向图D的边用Γ的元素标记。若有向环的标记有序乘积为Γ的单位元，则称该环为零和环。本文证明当d≥e³(k-1)时，必存在零和环；还证明每个带标记的d-正则有向图包含Ω(d/k)个两两顶点不交的零和环，当d≥50k时，包含Ω(d²/k)个两两边不交的零和环，所有三个结果均为渐近最优。存在性与 packing（ packing 指打包）结果可推广至欧拉有向图，其最小与最大公共度δ、Δ满足δ³/Δ²=Ω(k)，所用技术推广了Friedland针对偶有向环的行列式-永久式论证。
英文摘要
Let $Γ$ be a finite group of order $k\ge2$, and label the edges of a simple loopless $d$-regular digraph $D$ by elements of $Γ$. A directed cycle is zero-sum if the ordered product of its labels is the identity of $Γ$. We prove that a zero-sum cycle exists whenever $d\ge e^3(k-1)$. We also prove that every labelled $d$-regular digraph contains $Ω(d/k)$ pairwise vertex-disjoint zero-sum cycles. When $d\ge50k$, it contains $Ω(d^2/k)$ pairwise edge-disjoint zero-sum cycles. All three results are asymptotically optimal. The existence and packing results extend to Eulerian digraphs whose minimum and maximum common degrees $δ$ and $Δ$ satisfy $δ^3/Δ^2=Ω(k)$. The techniques extend a determinant--permanent argument of Friedland for even directed cycles.
