Nowhere-zero 3-flows in matroid base graph

Abstract

The base graph of a simple matroid M = (E,B) is the graph G such that V (G) = B and E(G) = {BB′: B,B′ ∈ B, |B / B′| = 1}, where the same notation is used for the vertices of G and the bases of M. It is proved that the base graph G of connected simple matroid M is Z3-connected if |V (G)| ⩽ 5. We also proved that if M is not a connected simple matroid, then the base graph G of M does not admit a nowhere-zero 3-flow if and only if |V (G)| = 4. Furthermore, if for every connected component Ei (i ⩽ 2) of M, the matroid base graph Gi of Mi = M|Ei has |V (Gi)| ⩽ 5, then G is Z3-connected which also implies that G admits nowhere-zero 3-flow immediately.