{"paper":{"title":"Optimal chromatic bound for ($P_3\\cup P_2$, house)-free graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Di Wu, Jinfeng Li, Rui Li","submitted_at":"2023-08-10T08:58:12Z","abstract_excerpt":"Let $G$ and $H$ be two vertex disjoint graphs. The {\\em union} $G\\cup H$ is the graph with $V(G\\cup H)=V(G)\\cup V(H)$ and $E(G\\cup H)=E(G)\\cup E(H)$. We use $P_k$ to denote a {\\em path} on $k$ vertices, use {\\em house} to denote the complement of $P_5$. In this paper, we show that $\\chi(G)\\le2\\omega(G)$ if $G$ is ($P_3\\cup P_2$, house)-free. Moreover, this bound is optimal when $\\omega(G)\\ge2$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2308.05442","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/2308.05442/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"}