Hall's Harem Theorem with controlled sizes of cycles

Abstract

We prove a new version of Hall's Harem Theorem, where the final matching is realized by a unary function with additional conditions on behavior of cycles. The present paper can be considered as a helpful companion of the paper of the author: arXiv:2105.06304, where a computable version of Hall's Harem Theorem with controlled sizes of cycles is proved. These two versions of Hall's Harem Theorem are independent: none of them follows from the other one.

This paper has overlap with arXiv:2105.06304v2. That version of mentioned preprint is splitted into two, one being this preprint, while the other concerns computable version of the theorem