Connected Fair Detachments of Hypergraphs

TL;DR

This paper establishes necessary and sufficient conditions for creating connected, fair detachments of hypergraphs with colored edges, solving a decades-old open problem and advancing hypergraph decomposition techniques.

Contribution

It provides the first complete characterization for fair detachments with connected color classes in hypergraphs, including the case of simple graphs.

Findings

Solved an open problem from the 1970s.

Established conditions for fair detachments with connected color classes.

Applied results to hypergraph decompositions and partial regular structures.

Abstract

Let $\mathcal G$ be a hypergraph whose edges are colored. An {\it $(\alpha,n)$-detachment} of $\mathcal G$ is a hypergraph obtained by splitting a vertex $\alpha$ into $n$ vertices, say $\alpha_1,\dots,\alpha_n$, and sharing the incident hinges and edges among the subvertices. A detachment is {\it 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 70s by finding necessary and sufficient conditions under which a $k$-edge-colored hypergraph $\mathcal G$ has a fair detachment in which each color class is connected. Previously, this was not even known for the case when $\mathcal G$ 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, and completing partial regular combinatorial structures.