Embedding connected factorizations

Bahmanian, Amin; Johnsen, Anna; Napirata, Stefan

doi:10.1016/j.jctb.2022.03.002

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2022

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“…His proof was based on an ingenious use of the Max-Flow Min-Cut Theorem. Generalizations and extensions of Baranyai's Theorem can be found in [3,4,5,6,8,9,18].…”

Section: Introductionmentioning

confidence: 99%

A non-uniform extension of Baranyai's Theorem

He¹,

Hao²,

Ma³

2022

Preprint

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A celebrated theorem of Baranyai states that when k divides n, the family K k n of all k-subsets of an n-element set can be partitioned into perfect matchings. In other words, K k n is 1-factorable. In this paper, we determine all n, k, such that the family K ≤k n consisting of subsets of [n] of size up to k is 1-factorable, and thus extend Baranyai's Theorem to the non-uniform setting. In particular, our result implies that for fixed k and sufficiently large n, K ≤k n is 1-factorable if and only if n ≡ 0 or −1 (mod k).

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“…His proof was based on an ingenious use of the Max-Flow Min-Cut Theorem. Generalizations and extensions of Baranyai's Theorem can be found in [3,4,5,6,8,9,18].…”

Section: Introductionmentioning

confidence: 99%