Uniquely 2-colourable 4-cycle decompositions
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916012664815616 |
|---|---|
| author | Burgess, Andrea C. Pike, David A. Pourakbar-Saffar, Shahriyar |
| author_facet | Burgess, Andrea C. Pike, David A. Pourakbar-Saffar, Shahriyar |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_14804 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uniquely 2-colourable 4-cycle decompositions Burgess, Andrea C. Pike, David A. Pourakbar-Saffar, Shahriyar Combinatorics 05C51, 05C15, 05B30 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. |
| title | Uniquely 2-colourable 4-cycle decompositions |
| topic | Combinatorics 05C51, 05C15, 05B30 |
| url | https://arxiv.org/abs/2605.14804 |