Colourings of Cayley graphs of finite $3$-groups

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 ${\mathcal{G}}_3(G)$ - a three-dimensional analogue of a Latin square - admits a proper colouring with $\mathrm{\mid }G\mathrm{\mid }$ 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.