Colourings of Uniform Group Divisible Designs and Maximum Packings

A weak $c$-colouring of a design is an assignment of colours to its points from a set of $c$ available colours, such that there are no monochromatic blocks. A colouring of a design is block-equitable, if for each block, the number of points coloured with any available pair of colours differ by at most one. Weak and block-equitable colourings of balanced incomplete block designs have been previously considered. In this paper, we extend these concepts to group divisible designs (GDDs) and packing designs. We first determine when a $k$-GDD of type $g^u$ can have a block-equitable $c$-colouring. We then give a direct construction of maximum block-equitable $2$-colourable packings with block size $4$; a recursive construction has previously appeared in the literature. We also generalise a bound given in the literature for the maximum size of block-equitably $2$-colourable packings to $c>2$. Furthermore, we establish the asymptotic existence of uniform $k$-GDDs with arbitrarily many groups and arbitrary chromatic numbers (with the exception of $c=2$ and $k=3$). A structural analysis of $2$- and $3$-uniform $3$-GDDs obtained from 4-chromatic STS$(v)$ where $v\in{21,25,27,33,37,39}$ is given. We briefly discuss weak colourings of packings, and finish by considering some further constraints on weak colourings of GDDs, namely requiring all groups to be either monochromatic or equitably coloured.

💡 Research Summary

The paper investigates colourings of two important classes of combinatorial designs – group divisible designs (GDDs) and packing designs (PDs) – extending concepts that have previously been studied mainly for balanced incomplete block designs (BIBDs).
After establishing precise definitions of weak colourings (no block is monochromatic) and block‑equitable colourings (in each block the number of points of any two colours differs by at most one), the authors first address block‑equitable colourings of uniform k‑GDDs of type gᵘ. The central result (Theorem 2.6) gives a complete characterisation: a k‑GDD(gᵘ) admits a block‑equitable c‑colouring if and only if one of three conditions holds – (i) u ≤ c ≤ ug, (ii) k = u, or (iii) k = u − 1 and c divides u. The proof combines a counting argument on monochromatic pairs with explicit constructions, showing that the optimal way to minimise monochromatic pairs is to colour each group monochromatically.
The paper then turns to maximum packings PD(v,k,1) that admit block‑equitable colourings. Building on a bound for the case c = 2 found in earlier work, the authors generalise the upper bound to any c > 2. They prove that the bound is tight whenever c divides k and a transversal design TD(k, v/k) exists; a direct construction is given that distributes the colours cyclically across blocks so that each colour pair appears almost equally often.
Section 3 focuses on weak colourings. After recalling known results for BIBDs, the authors prove (Corollary 3.9) that for any admissible parameters (k,λ) and for any number of colours c (except the special case c = 2, k = 3) there exist uniform k‑GDDs with arbitrarily many groups that are weakly c‑colourable. The proof uses recursive constructions based on transversal designs and group‑inflation techniques, establishing asymptotic existence. A detailed structural analysis of 2‑ and 3‑uniform 3‑GDDs derived from 4‑chromatic Steiner triple systems with v∈{21,25,27,33,37,39} is also presented, illustrating how the colour classes can be arranged to satisfy both weak and block‑equitable requirements.
Section 4 adds further constraints on weak colourings: (i) all groups must be monochromatic, and (ii) all groups must be point‑equitable. The authors derive necessary congruence conditions and prove existence results under these stricter regimes. In particular, Corollary 4.10 shows that when u ≥ 4, (u − 1)g≡0 (mod 3) and u(u − 1)g²≡0 (mod 12), there exists a 2‑chromatic 4‑GDD of type (4g)ᵘ whose groups are point‑equitable. Similar extensions are obtained for larger block sizes when appropriate arithmetic conditions are satisfied.
The paper concludes by summarising the contributions: a full characterisation of block‑equitable colourings for uniform GDDs, a general bound and constructive realisation for block‑equitable packings, asymptotic existence of weakly colourable GDDs with arbitrary chromatic numbers, and new results under group‑level colour constraints. These findings broaden the theory of colourings in combinatorial design, linking it with number‑theoretic conditions and providing constructions that may be useful in applications such as DNA‑based data storage, network design, and error‑correcting codes.