{"paper":{"title":"On odd covers of cliques and disjoint unions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexander Clifton, Calum Buchanan, Eric Culver, Jiaxi Nie, Kenta Ozeki, Mei Yin, P\\'eter Frankl, Puck Rombach","submitted_at":"2024-08-16T08:20:11Z","abstract_excerpt":"Babai and Frankl posed the ``odd cover problem\" of finding the minimum cardinality of a collection of complete bipartite graphs such that every edge of the complete graph of order $n$ is covered an odd number of times. In a previous paper with O'Neill, some of the authors proved that this value is always $\\lceil n / 2 \\rceil$ or $\\lceil n / 2 \\rceil + 1$ and that it is the former whenever $n$ is a multiple of $8$. In this paper, we determine this value to be $\\lceil n / 2 \\rceil$ whenever $n$ is odd or equivalent to $18$ modulo $24$. We also further the study of odd covers of graphs which are "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.08598","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2408.08598/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"}