Colourings of Uniform Group Divisible Designs and Maximum Packings

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 $\left(v\right)$, where $v\in \left\{21,25,27,33,37,39\right\}$ 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.