{"paper":{"title":"On interval colourings of graphs","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Julien Portier, Lawrence Hollom, Leo Versteegen","submitted_at":"2023-03-09T18:56:26Z","abstract_excerpt":"An interval colouring of a graph $G=(V,E)$ is a proper colouring $c\\colon E\\to \\mathbb{Z}$ such that the set of colours of edges incident to any given vertex forms an interval of $\\mathbb{Z}$. The interval thickness $\\theta(G)$ of a graph $G$ is the smallest integer $k$ such that $G$ can be edge-partitioned into $k$ interval colourable graphs, and $\\theta(n)$ is the largest interval thickness over graphs on $n$ vertices. We show that $c \\frac{\\log n}{\\log \\log n} \\leq \\theta(n) \\leq n^{8/9+o(1)}$ for some $c>0$. In particular this answers a question by Asratian, Casselgren, and Petrosyan.\n  In"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.05505","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2303.05505/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}