Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Parity Systems and the Delta-Matroid Intersection Problem. | Parity Systems and the Delta-Matroid Intersection Problem Andre Bouchet and Bill Jackson 1 Submitted February 16 1998 Accepted September 3 1999. Abstract 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 Lovasz to a new abstract system which we call a parity system. Examples of parity systems may be obtained by combining either two delta-matroids or two orthogonal 2-polymatroids on the same ground-sets. We show that many of the results of Lovasz concerning double flowers and projections carry over to parity systems. 1 Introduction the delta-matroid intersection problem A delta-matroid is a pair V B with a finite set V and a nonempty collection B of subsets of V called the feasible sets or bases satisfying the following axiom Departement d informatique Universite du Maine 72017 Le Mans Cedex France. bouchet@ Department of Mathematical and Computing Sciences Goldsmiths College London SE14 6NW England. maa01wj@ 1 THE ELECTRONIC .JOURNAL OF COMBINATORICS 7 2000 R14 2 For Bl and B2 in B and V1 in B-Ị-AB2 there is v2 in AB2 such that B1A v1 v2 belongs to B. Here PAQ P Q u Q P is the symmetric difference of two subsets P and Q of V. If X is a subset of V and if we set BAX BAX B E B then we note that V BAX is a new delta-matroid. The transformation V B V BAX is called a twisting. The rank of a subset P of V is r P max P n B V P n V B . 1 Thus r P V if and only if P belongs to B. Proposition A nonempty collection B of subsets of V is the collection of bases of a matroid if and only if V B is a delta-matroid and the members of B have the same cardinality. We refer the reader to 4 for an introduction to delta-matroids and some of the problems considered in that paper. Problem Given delta-matroids V B1 and V B2 with rank functions r1 and r2 respectively search for a subset P of V that maximizes r1 P r2
