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Article Dans Une Revue SIAM Journal on Discrete Mathematics Année : 2018

Additive bases and flows in graphs

Résumé

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.

Dates et versions

hal-01709239 , version 1 (14-02-2018)

Identifiants

Citer

Louis Esperet, Rémi de Joannis de Verclos, Tien-Nam Le, Stéphan Thomassé. Additive bases and flows in graphs. SIAM Journal on Discrete Mathematics, 2018, 32 (1), pp.534-542. ⟨10.1137/17M1113758⟩. ⟨hal-01709239⟩
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