{"paper":{"title":"A tight linear chromatic bound for ($P_3\\cup P_2, W_4$)-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":"2023-08-17T03:51:53Z","abstract_excerpt":"For two vertex disjoint graphs $H$ and $F$, we use $H\\cup F$ to denote the graph with vertex set $V(H)\\cup V(F)$ and edge set $E(H)\\cup E(F)$, and use $H+F$ to denote the graph with vertex set $V(H)\\cup V(F)$ and edge set $E(H)\\cup E(F)\\cup\\{xy\\;|\\; x\\in V(H), y\\in V(F)$$\\}$. A $W_4$ is the graph $K_1+C_4$. In this paper, we prove that $\\chi(G)\\le 2\\omega(G)$ if $G$ is a ($P_3\\cup P_2, W_4$)-free graph. This bound is tight when $\\omega =2$ and $3$, and improves the main result of Wang and Zhang. Also, this bound partially generalizes some results of Prashant {\\em et al.}."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2308.08768","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/2308.08768/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"}