{"paper":{"title":"Nearly optimal coloring of some C4-free graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Baogang Xu, Ran Chen","submitted_at":"2024-09-11T01:55:23Z","abstract_excerpt":"A class ${\\cal G}$ of graphs is $\\chi$-{\\em polydet} if ${\\cal G}$ has a polynomial binding function $f$ and there is a polynomial time algorithm to determine an $f(\\omega(G))$-coloring of $G\\in {\\cal G}$. Let $P_t$ and $C_t$ denote a path and a cycle on $t$ vertices, respectively. A {\\em bull} consists of a triangle with two disjoint pendant edges, a {\\em hammer} is obtained by identifying an end of $P_3$ with a vertex of a triangle, a {\\em fork$^+$} is obtained from $K_{1, 3}$ by subdividing an edge twice. Let $H$ be a bull or a hammer, and $F$ be a $P_7$ or a fork$^+$. We determine all $(C_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.06944","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/2409.06944/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"}