Rainbow Fractional Matchings

Abstract

We prove that any family E 1 , . . . , E ⌈rn⌉ of (not necessarily distinct) sets of edges in an r-uniform hypergraph, each having a fractional matching of size n, has a rainbow fractional matching of size n (that is, a set of edges from distinct E i 's which supports such a fractional matching). When the hypergraph is r-partite and n is an integer, the number of sets needed goes down from rn to rn − r + 1. The problem solved here is a fractional version of the corresponding problem about rainbow matchings, which was solved by Drisko and by Aharoni and Berger in the case of bipartite graphs, but is open for general graphs as well as for r-partite hypergraphs with r > 2. Our topological proof is based on a result of Kalai and Meshulam about a simplicial complex and a matroid on the same vertex set.

Choice Functions in the Intersection of Matroids

“…In fact, this is a special case of the main theorem in [8]. The way to derive it from the original theorem can be found in [3].…”

“…We shall use the acronym PDS for "positive, decreasing and submodular". As in [3], we shall consider perturbations of X k . For this purpose, we shall need the following:…”

“…Following a crucial idea from [3], we may assume that generically, for every W ⊆ V there is a unique function h on [r] × 2 V attaining the minimum in the program defining τ * a,b (W ). For, the set of all b = (b 1 , .…”

“…In [2] it was conjectured that almost the same is true in general graphs, namely that in any graph 2k matchings of size k have a rainbow matching of size k, and that for odd k the Drisko bound suffices -2k − 1 matchings of size k have a rainbow matching of size k. This is far from being solved (in [2] the bound 3k − 2 was proved), but in [3] a fractional version of the conjecture was proved, Date: July 4, 2019. in a more general setting. Recall that ν * (F ) denotes the largest total weight of a fractional matching in a hypergraph H. Theorem 1.2 (Aharoni, Holzman and Jiang [3]). Let m be a real number, let H be an r-uniform hypergraph and let q ≥ ⌈rk⌉ be an integer.…”

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We prove a common generalization of two results, one on rainbow fractional matchings [3] and one on rainbow sets in the intersection of two matroids [9]: Given d = r⌈k⌉−r +1 functions of size (=sum of values) k that are all independent in each of r given matroids, there exists a rainbow set of supp(f i ), i ≤ d, supporting a function with the same properties.

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