Colourings of Uniform Group Divisible Designs and Maximum Packings
Andrea C Burgess, Peter Danziger, Diane Donovan, Tara Kemp, James G. Lefevre, David A. Pike, E. Sule Yazici
TL;DR
The paper addresses colourings of block designs, focusing on block-equitable and weak colourings for BIBDs, GDDs, and packings. It provides a complete characterization of when a $k$-GDD of type $g^u$ admits a block-equitable colouration, offers a direct construction and size bounds for maximum block-equitable colourable packings, and proves asymptotic existence results for uniform $k$-GDDs with arbitrary chromatic numbers (excluding the case $(c,k)=(2,3)$). It also develops weak-colouring theory for GDDs and packings, and introduces monochromatic-group and group-equitable colourings with constructive results via transversal designs and blow-ups, including new existence results for group-equitable $2$-colourings of certain $4$-GDDs. Collectively, these results connect colouring constraints to explicit design constructions with potential applications in experimental design, coding, and data encoding schemes.
Abstract
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.