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arXiv · 2605.14804

Uniquely 2-colourable 4-cycle decompositions

Andrea C. Burgess·David A. Pike·Shahriyar Pourakbar-Saffar
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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.

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Andrea C. Burgess, David A. Pike, Shahriyar Pourakbar-Saffar. 2026-05-14. Uniquely 2-colourable 4-cycle decompositions. https://arxiv.org/abs/2605.14804

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