Additive bases and flows in graphs

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 Zpn forms an additive basis of Zpn (i.e. any element of Zpn 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⩾3 is a prime number and G→ is a directed, highly edge-connected graph in which each arc is given a list of two distinct values in Zp. Then G→ has a Zp-flow in which each arc is assigned a value of its own list.