Matroid matching with Dilworth truncation

Abstract

Let H = ( V , E ) be a hypergraph and let k ⩾ 1 and l ⩾ 0 be fixed integers. Let M be the matroid with ground-set E s.t. a set F ⊆ E is independent if and only if each X ⊆ V with k | X | - l ⩾ 0 spans at most k | X | - l hyperedges of F. We prove that if H is dense enough, then M satisfies the double circuit property, thus Lovász’ min–max formula on the maximum matroid matching holds for M . Our result implies the Berge–Tutte formula on the maximum matching of graphs ( k = 1 , l = 0 ), generalizes Lovász’ graphic matroid (cycle matroid) matching formula to hypergraphs ( k = l = 1 ) and gives a min–max formula for the maximum matroid matching in the two-dimensional rigidity matroid ( k = 2 , l = 3 ).