{"paper":{"title":"Graph tilings in incompatibility systems","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Donglei Yang, Hao Li, Jie Hu, Yue Wang","submitted_at":"2022-07-12T08:32:26Z","abstract_excerpt":"An \\emph{incompatibility system} $(G,\\mathcal{F})$ consists of a graph $G$ and a family $\\mathcal{F}=\\{F_v\\}_{v\\in V(G)}$ over $G$ with $F_v\\subseteq \\{\\{e,e'\\}\\in {E(G)\\choose 2}: e\\cap e'=\\{v\\}\\}$. We say that two edges $e,e'\\in E(G)$ are \\emph{incompatible} if $\\{e,e'\\}\\in F_v$ for some $v\\in V(G)$, and otherwise \\emph{compatible}. A subgraph $H$ of $G$ is \\emph{compatible} if every pair of edges in $H$ are compatible. An incompatibility system $(G,\\mathcal{F})$ is \\emph{$\\Delta$-bounded} if for any vertex $v$ and any edge $e$ incident with $v$, there are at most $\\Delta$ members of $F_v$ c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2207.05386","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/2207.05386/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"}