Colourings of Cayley graphs of finite $3$-groups
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866915887891611648 |
|---|---|
| author | Grzeszczuk, Piotr |
| author_facet | Grzeszczuk, Piotr |
| contents | Colouring problems arising from group-based constructions provide a natural link between combinatorics and algebra, particularly in the study of Cayley graphs and Latin squares. We introduce the notion of colouring bijections of finite groups, a class of permutations encoding proper vertex colourings of associated Cayley-type graphs, extending classical concepts such as complete mappings and strong complete mappings.
We prove that every finite $3$-group without a cyclic maximal subgroup admits a colouring bijection. Consequently, for such a group $G$, the graph $\mathscr{G}_3(G)$ - a three-dimensional analogue of a Latin square - admits a proper colouring with $|G|$ colours.
These results show that the existence of colouring bijections is governed by structural properties of $3$-groups, revealing a new connection between group theory and combinatorial colouring problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_20084 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Colourings of Cayley graphs of finite $3$-groups Grzeszczuk, Piotr Combinatorics Group Theory 05C15, 20D15, 05C25 Colouring problems arising from group-based constructions provide a natural link between combinatorics and algebra, particularly in the study of Cayley graphs and Latin squares. We introduce the notion of colouring bijections of finite groups, a class of permutations encoding proper vertex colourings of associated Cayley-type graphs, extending classical concepts such as complete mappings and strong complete mappings. We prove that every finite $3$-group without a cyclic maximal subgroup admits a colouring bijection. Consequently, for such a group $G$, the graph $\mathscr{G}_3(G)$ - a three-dimensional analogue of a Latin square - admits a proper colouring with $|G|$ colours. These results show that the existence of colouring bijections is governed by structural properties of $3$-groups, revealing a new connection between group theory and combinatorial colouring problems. |
| title | Colourings of Cayley graphs of finite $3$-groups |
| topic | Combinatorics Group Theory 05C15, 20D15, 05C25 |
| url | https://arxiv.org/abs/2603.20084 |