Document Type

Article

Publication Title

Combinatorica

Publication Date

2026

Keywords

Baranyai's theorem, detachment, amalgamation, factorization, connectivity

Abstract

Let G be a hypergraph whose edges are colored. An (α, n)-detachment of G is a hypergraph obtained by splitting a vertex α into n vertices, say α1, . . . , αn , and sharing the incident edges among the subvertices. A detachment is fair if the degree of vertices and multiplicity of edges are shared as evenly as possible among the subvertices within the whole hypergraph as well as within each color class. In this paper we solve an open problem from the 1970s by finding necessary and sufficient conditions under which a k-edge-colored hypergraph G has a fair detachment in which each color class is connected. Previously, this was not even known for the case when G is an arbitrary graph (i.e. 2-uniform hypergraph). We exhibit the usefulness of our theorem by proving a variety of new results on hypergraph decompositions (in Part I), and completing partial regular combinatorial structures (in Part II)

Funding Source

The author’s research is partially supported by NSA Grant H98230-16-1-0304. This article was published Open Access thanks to a transformative agreement between Milner Library and Springer Nature.

Comments

First published in Combinatorica (2026): https://doi.org/10.1007/s00493-026-00224-z

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

DOI

10.1007/s00493-026-00224-z

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