Colourings of Cayley graphs of finite $3$-groups

TL;DR

This paper introduces colouring bijections for finite groups, linking group structure to graph colourings, and proves their existence for certain 3-groups, thus connecting algebraic properties with combinatorial colouring problems.

Contribution

It defines colouring bijections for finite groups and proves their existence for 3-groups without cyclic maximal subgroups, revealing new algebraic-combinatorial connections.

Findings

Existence of colouring bijections for specific 3-groups

Proper colourings of Cayley graphs with |G| colours

Structural properties of 3-groups govern colouring bijections

Abstract

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.