The positive circuits of oriented matroids with the packing property or idealness

The class of clutters of the positive circuits of oriented matroids is an important class which generalizes both the dicut clutters and the dicycle clutters of directed graphs, together. In this article, we will show that the clutter of the positive circuits of an oriented matroid whose co-rank is at most 4 has the packing property if and only if it has none of the six minimally non-packing minors J 3 , C 3 2 , C 5 3 , Q 6 , Q 6 ⊗ 1 and Q 6 ⊗ { 1 , 2 } . By using this, we will also prove that the clutter of the positive circuits of an oriented matroid whose co-rank is at most 4 is ideal if and only if it has none of the three minimally non-packing minors J 3 , C 3 2 , and C 5 3 .