Uniquely 2-colourable 4-cycle decompositions

TL;DR

This paper constructs uniquely 2-colourable 4-cycle decompositions of complete graphs for all admissible orders above certain thresholds, advancing understanding of colourability in cycle systems.

Contribution

It provides explicit constructions of uniquely 2-colourable 4-cycle systems for all admissible orders n ≥ 49, expanding the class of known such systems.

Findings

Constructed uniquely 2-colourable 4-cycle systems for all n ≥ 49.

Extended these constructions to decompositions of K_n minus an edge for all n ≥ 50.

Contributed new examples and methods to the study of colourability in cycle systems.

Abstract

A cycle system of order $n$ is a decomposition of the edges of the complete graph $K_n$ into cycles of a fixed length. A cycle system is said to be $k$-colourable if we can assign $k$ colours to its vertices so that no cycle is monochromatic. A $k$-colourable cycle system is uniquely $k$-colourable if its colouring is unique up to the permutation of colour classes. In this paper, we construct uniquely $2$-colourable $4$-cycle systems of order $n$ for all admissible $n\geq 49$, and also uniquely $2$-colourable $4$-cycle decompositions of $K_n - I$, for all admissible $n \geq 50$. These constructions contribute to the broader study of uniquely colourable cycle systems and open new directions for future research.