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A Fractional Analogue of Brooks' Theorem

Let $Δ(G)$ be the maximum degree of a graph $G$. Brooks' theorem states that the only connected graphs with chromatic number $χ(G)=Δ(G)+1$ 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 $χ_f(G)$ is at least $Δ(G)$. These graphs are complete graphs, odd cycles, $C^2_8$, $C_5\boxtimes K_2$, and graphs whose clique number $ω(G)$ equals the maximum degree $Δ(G)$. Among the two sporadic graphs, the graph $C^2_8$ is the square graph of cycle $C_8$ while the other graph $C_5\boxtimes K_2$ is the strong product of $C_5$ and $K_2$. In fact, we prove a stronger result; if a connected graph $G$ with $Δ(G)\geq 4$ is not one of the graphs listed above, then we have $χ_f(G)\leq Δ(G)- 2/67$.