图的可加基与可加流
Additive bases and flows in graphs
摘要：
Jaeger、Linial、Payan和Tarsi在1992年证明，对于任何素数$p$，存在一个常数$c$，使得对于任何$n$，$\mathbb{Z}_p^n$的任何$c$线性基族的向量的并集（具有重复）形成$\mathbb{Z}_p^n$的加法基（即$\mathbb{Z}_p^n$的任何元素都可以表示为这些向量的子集的和）。在本文中，我们证明了这个猜想时，每个向量包含最多两个非零元素。作为应用，我们证明了关于高边连通图中流的几个结果，推广了已知的结果.例如，假设$p\ge 3$是一个素数，$\vec{G}$是一个有向的、高度边连通图，其中每个弧在$\mathbb{Z}_p$中被给予一个由两个不同值组成的列表。然后$\vec{G}$有一个$\mathbb{Z}_p$流，其中每个弧被分配一个它自己列表的值。
英文摘要：
It was conjectured by Jaeger, Linial, Payan, and Tarsi in 1992 that for any prime number $p$, there is a constant $c$ such that for any $n$, the union (with repetition) of the vectors of any family of $c$ linear bases of $\mathbb{Z}_p^n$ forms an additive basis of $\mathbb{Z}_p^n$ (i.e. any element of $\mathbb{Z}_p^n$ can be expressed as the sum of a subset of these vectors). In this note, we prove this conjecture when each vector contains at most two non-zero entries. As an application, we prove several results on flows in highly edge-connected graphs, extending known results. For instance, assume that $p\ge 3$ is a prime number and $\vec{G}$ is a directed, highly edge-connected graph in which each arc is given a list of two distinct values in $\mathbb{Z}_p$. Then $\vec{G}$ has a $\mathbb{Z}_p$-flow in which each arc is assigned a value of its own list.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
