{"paper":{"title":"Uniquely 2-colourable 4-cycle decompositions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrea C. Burgess, David A. Pike, Shahriyar Pourakbar-Saffar","submitted_at":"2026-05-14T13:14:32Z","abstract_excerpt":"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 construc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.14804","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}