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 gu 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 vE{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.