Additive bases and flows in graphs

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.

14 pages, no figure