Characterizing matchings as the intersection of matroids

This paper deals with the problem of representing the matching independence system in a graph as the intersection of finitely many matroids. After characterizing the graphs for which the matching independence system is the intersection of two matroids, we study the function mu(G), which is the minimum number of matroids that need to be intersected in order to obtain the set of matchings on a graph G, and examine the maximal value, mu(n), for graphs with n vertices. We describe an integer programming formulation for deciding whether mu(G) less than or equal to k. Using combinatorial arguments, we prove that mu(n) is an element of Omegal(log log n). On the other hand, we establish that mu(n) is an element of O(log n/log log n). Finally, we prove that mu(n) = 4 for n = 5,...,12, and sketch a proof of mu(n) = 5 for n = 13, 14, 15.