, Volume 58, Issue 2, pp 319-329

Characterizing matchings as the intersection of matroids

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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 μ(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, μ(n), for graphs with n vertices. We describe an integer programming formulation for deciding whether μ(G)≤k. Using combinatorial arguments, we prove that μ(n)∈Ω(log logn). On the other hand, we establish that μ(n)∈O(logn/ log logn). Finally, we prove that μ(n)=4 for n=5,…,12, and sketch a proof of μ(n)=5 for n=13,14,15.

An earlier version appears as an extended abstract in the Proceedings of COMB’01 [5]. Supported by the “Gerhard-Hess-Forschungs-Förderpreis” (WE 1462) of the German Science Foundation (DFG) awarded to R. Weismantel.