{"paper":{"title":"Equitable block colourings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Lucia Marino, Paola Bonacini","submitted_at":"2014-06-20T16:29:07Z","abstract_excerpt":"Let $\\Sigma=(X,\\mathcal B)$ a $4$-cycle system of order $v=1+8k$. A $c$-colouring of type $s$ is a map $\\phi\\colon \\mathcal B\\rightarrow \\mathcal C$, with $C$ set of colours, such that exactly $c$ colours are used and for every vertex $x$ all the blocks containing $x$ are coloured exactly with $s$ colours. Let $4k=qs+r$, with $q,r\\ge 0$. $\\phi$ is \\emph{equitable} if for every vertex $x$ the set of the $4k$ blocks containing $x$ is parted in $r$ colour classes of cardinality $q+1$ and $s-r$ colour classes of cardinality $q$. In this paper we study colourings for which $s|k$, giving a descripti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.5454","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"}