The $h$-vectors of 1-dimensional Matroid Complexes and a Conjecture of Stanley

A matroid complex is a pure complex such that every restriction is again pure. It is a long-standing open problem to classify all possible $h$-vectors of such complexes. In the case when the complex has dimension 1 we completely resolve this question. We also prove the 1-dimensional case of a conjecture of Stanley that all matroid $h$-vectors are pure $O$-sequences. Finally, we completely characterize the Stanley-Reisner ideals of 1-dimensional matroid complexes.