# Additive bases and flows in graphs

## TL;DR

This paper proves a conjecture about additive bases in finite vector spaces for vectors with at most two non-zero entries and applies this to establish new flow results in highly edge-connected graphs.

## Contribution

It confirms the conjecture for vectors with limited non-zero entries and extends flow existence results in highly edge-connected graphs.

## Key findings

- Confirmed the conjecture for vectors with at most two non-zero entries.
- Established new flow existence results in highly edge-connected graphs.
- Extended known results on flows in directed graphs with list constraints.

## Abstract

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.