Parity Systems and the Delta-Matroid Intersection Problem

We consider the problem of determining when two delta-matroids on the same ground-set have a common base. Our approach is to adapt the theory of matchings in 2-polymatroids developed by Lovász to a new abstract system, which we call a parity system. Examples of parity systems may be obtained by combining either, two deltamatroids, or two orthogonal 2-polymatroids, on the same ground-sets. We show that many of the results of Lovász concerning ‘double flowers’ and ‘projections’ carry over to parity systems.