Connected block decomposition with the natural edge threshold

Construct, for every d≥2 and every positive integer λ, a coloring satisfying the hypotheses and balance conditions of Theorem 2.1 such that each color subhypergraph G^(j) is connected whenever m_j≥⌈(dn−1)/(d−1)⌉.

Background

Theorem 2.1 produces a coloring of the λ-fold complete d-uniform d-partite hypergraph with a prescribed block partition, simultaneously controlling color-class sizes, vertex degrees, block counts, and edge multiplicities.

A connected spanning subhypergraph on dn vertices with edges of size d must have at least ⌈(dn−1)/(d−1)⌉ edges. The authors propose that this necessary edge-count condition is also sufficient for choosing the decomposition so that every sufficiently large color class induces a connected spanning subhypergraph.

References

Finally, a connected version of Baranyai’s theorem [4] suggests a connected analogue of Theo- rem 2.1. Since Kd nˆd has dn vertices and its edges have size d, a connected spanning subhypergraph must have at least R dn ´ 1d ´ 1Vedges. We conjecture that this necessary condition is also sufficient.

— Sudoku Analogues of Baranyai's Theorem  (2609.23975 - Bahmanian et al., 21 Sep 2026) in Conjecture 5.3, Section 5, page 15