Excluded Minors of Rank 3 for Orientability and Representability

抄録

We investigate excluded minor characterizations of two fundamental classes of matroids: orientable matroids and representable matroids. We prove (i) for any fixed field F, there exist infinitely many excluded minors of rank 3 for the union of the class of orientable matroids and the class of F-representable matroids, and (ii) for any fixed field F with characteristic 0, there exist infinitely many orientable excluded minors of rank 3 for intersection of the class of orientable matroids and the class of F-representable matroids. We show these statements by explicitly constructing infinite families of excluded minors.