Cooperative Conditions for the Existence of Rainbow Matchings

Abstract

Let $k>1$, and let $\mathcal{F}$ be a family of $2n+k-3$ non-empty sets of edges in a bipartite graph. If the union of every $k$ members of $\mathcal{F}$ contains a matching of size $n$, then there exists an $\mathcal{F}$-rainbow matching of size $n$. Replacing $2n+k-3$ by $2n+k-2$, the result is true also for $k=1$, and it can be proved (for all $k$) both topologically and by a relatively simple combinatorial argument. The main effort is in gaining the last $1$, which makes the result sharp.

A proof of Frankl's conjecture on cross-union families

“…While the maximum of the geometric mean and the arithmetic mean of the families satisfying U ps`1, kps`1q´1q may behave differently from what is conjectured in the Erdős matching conjecture, it has been conjectured [1,17] that the minimum size behaves as in the Erdős matching conjecture. Recently Kupavskii [18] proved that this conjecture when s is sufficiently large and n ą 3eps`1qk.…”

On the rainbow matching conjecture for 3-uniform hypergraphs

“…The following analogous conjecture, known as the rainbow matching conjecture, was made by Aharoni and Howard [1] and, independently, by Huang, Loh, and Sudakov [13]. For related topics on rainbow type problems, we refer the interested reader to [14,16,18,21].…”

“…It was known that this conjecture holds for the case k = 2 (see a proof by Meshulam in [1]). The following result was obtained by Huang, Loh, and Sudakov [13].…”

Adaptive hypergraph superpixels

“…[27] Let 1/t ≪ γ ≪ µ, suppose that L 1 and L 2 are graphs on a common vertex set of size t such that L 1 , L 2 has edge density at least 5/9 + µ. For i ∈ [2], let C i be a tight component of L i with a maximum number of edges. We have (i) C 1 and C 2 has an edge in common, (ii) C i has a switcher for i ∈ [2], (iii) C i has a fractional matching of density 1/3 + γ for i ∈ [2], (iv) C i has edge density at least 4/9 + γ for i ∈ [2].…”

“…For i ∈ [2], let C i be a tight component of L i with a maximum number of edges. We have (i) C 1 and C 2 has an edge in common, (ii) C i has a switcher for i ∈ [2], (iii) C i has a fractional matching of density 1/3 + γ for i ∈ [2], (iv) C i has edge density at least 4/9 + γ for i ∈ [2].…”