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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 $\\Delta(G)$. 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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 $\\Delta(G)$. 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