Nowhere-zero 3-flows in matroid base graph.

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 Z-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 E ( i ⩽ 2) of M, the matroid base graph G of M = M| E has | V ( G)| ⩽ 5, then G is Z-connected which also implies that G admits nowhere-zero 3-flow immediately. [ABSTRACT FROM AUTHOR]