Note on a matroid with parity conditon

This paper presents a theorem concerning a matroid with the parity condition. The theorem provides matroid generalizations of graph-theoretic results of Lewin [3] and Gallai [1]. Let M=(E, F) be a matroid, where E is a finite set of elements and F is the family of independent sets of M (in this paper, we presuppose a knowledge of matroid theory; our standard reference is Welsh [4]). Assume that E is of even cardinality and let P be a partition of E into disjoint blocks, each of size 2. A triple M = (E, F, P) is called a matroid with the parity condition. One of two elements of each block of P is called the mate of the other. The mate of eEE is denoted by ~. A set X ~ E is called a parity set, if for each eEX, ~EX. The matroid parity problem is to find a maximum independent parity set (an independent parity set with maximum number of elements). It is known that the matching problem of graphs is a special case of the matroid parity problem [2]. The purpose of this paper is to present a theorem on the maximal independent parity set of matroids with parity conditions. It is shown that the theorem provides matroid generalizations of results on the maximum matching of graphs by Lewin [3] and Gallai [1].