Connected Fair Detachments of Hypergraphs II

Source abstract

We study embeddings of factorizations of the $λ$-fold complete $h$-uniform hypergraph $λK_m^h$ in factorizations of $λK_n^h$, where $m<n$ and both the source and target degrees may vary by color. From every embedding one can obtain another that minimizes the number of components in every target color without increasing its multiplicity spread. This change also does not increase the sum of any convex function of the edge multiplicities, simultaneously in all colors. For $h=2,3$ we give exact existence criteria for every $m<n$. For $h\ge4$, the necessary divisibility and degree-sum conditions are sufficient once $n\ge(h-1)m$, improving the previously known threshold $n\ge hm$ for color-dependent degrees. In this range we also give exact criteria for connected, simple, and equimultiple target factors. The proofs use a system of integer counts recording the number of added edges of each type. Every system satisfying these equations gives an embedding with the minimum multiplicity spread allowed by