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Colourings of Cayley graphs of finite $3$-groups

Piotr Grzeszczuk

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 colouring bijections of finite groups, a class of permutations encoding proper vertex colourings of associated Cayley-type graphs, extending classical notions such as complete and strong complete mappings. We prove that every finite $3$-group without a cyclic maximal subgroup admits a colouring bijection. Consequently, for such groups $G$, the graph $\mathscr{G}_3(G)$ 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.

Colourings of Cayley graphs of finite $3$-groups

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 colouring bijections of finite groups, a class of permutations encoding proper vertex colourings of associated Cayley-type graphs, extending classical notions such as complete and strong complete mappings. We prove that every finite -group without a cyclic maximal subgroup admits a colouring bijection. Consequently, for such groups , the graph admits a proper colouring with colours. These results show that the existence of colouring bijections is governed by structural properties of -groups, revealing a new connection between group theory and combinatorial colouring problems.
Paper Structure (8 sections, 16 theorems, 150 equations)

This paper contains 8 sections, 16 theorems, 150 equations.

Table of Contents

  1. Preliminary results
  2. Colourability of groups of small order
  3. Lifting colouring bijections
  4. Case H = C3 x C3
  5. Case H = C3 x C9
  6. Main theorem
  7. Bijections for Z9 x Z3
  8. Tables for Lemma \ref{['C9C3-bijections']}

Key Result

Proposition 1.1

Suppose that a group $G$ admits a colouring bijection $\sigma\colon G\to G$. Then the function $c\colon G^3\to G$ defined by is an $|G|$-colouring of the graph $\mathscr{G}_3(G)$.

Theorems & Definitions (31)

  • Proposition 1.1
  • proof
  • Corollary 1.2
  • proof
  • Theorem 1.3
  • Lemma 1.4: Layer criterion
  • proof
  • Lemma 1.5
  • proof
  • Lemma 1.6
  • ...and 21 more