{"paper":{"title":"Clique covers and decompositions of cliques of graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jialin He, J\\'ozsef Balogh, Michael C. Wigal, Robert A. Krueger, The Nguyen","submitted_at":"2024-12-07T03:37:42Z","abstract_excerpt":"In 1966, Erd\\H{o}s, Goodman, and P\\'{o}sa showed that if $G$ is an $n$-vertex graph, then at most $\\lfloor n^2/4 \\rfloor$ cliques of $G$ are needed to cover the edges of $G$, and the bound is best possible as witnessed by the balanced complete bipartite graph. This was generalized independently by Gy\\H{o}ri--Kostochka, Kahn, and Chung, who showed that every $n$-vertex graph admits an edge-decomposition into cliques of total `cost' at most $2 \\lfloor n^2/4 \\rfloor$, where an $i$-vertex clique has cost $i$. Erd\\H{o}s suggested the following strengthening: every $n$-vertex graph admits an edge-de"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.05522","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/2412.05522/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"}