Colourings of Uniform Group Divisible Designs and Maximum Packings
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911459723706368 |
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| author | Burgess, Andrea C Danziger, Peter Donovan, Diane Kemp, Tara Lefevre, James G. Pike, David A. Yazıcı, E. Şule |
| author_facet | Burgess, Andrea C Danziger, Peter Donovan, Diane Kemp, Tara Lefevre, James G. Pike, David A. Yazıcı, E. Şule |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_05375 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Colourings of Uniform Group Divisible Designs and Maximum Packings Burgess, Andrea C Danziger, Peter Donovan, Diane Kemp, Tara Lefevre, James G. Pike, David A. Yazıcı, E. Şule Combinatorics 05B05, 05B30, 05B40, 05C15 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. |
| title | Colourings of Uniform Group Divisible Designs and Maximum Packings |
| topic | Combinatorics 05B05, 05B30, 05B40, 05C15 |
| url | https://arxiv.org/abs/2510.05375 |