{"paper":{"title":"A Fractional Analogue of Brooks' Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrew D. King, Linyuan Lu, Xing Peng","submitted_at":"2011-03-17T21:10:53Z","abstract_excerpt":"Let $\\Delta(G)$ be the maximum degree of a graph $G$. Brooks' theorem states that the only connected graphs with chromatic number $\\chi(G)=\\Delta(G)+1$ are complete graphs and odd cycles. We prove a fractional analogue of Brooks' theorem in this paper. Namely, we classify all connected graphs $G$ such that the fractional chromatic number $\\chi_f(G)$ is at least $\\Delta(G)$. These graphs are complete graphs, odd cycles, $C^2_8$, $C_5\\boxtimes K_2$, and graphs whose clique number $\\omega(G)$ equals the maximum degree $\\Delta(G)$. Among the two sporadic graphs, the graph $C^2_8$ is the square gra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.3524","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}