The fractional chromatic number of $K_{\Delta}$-free graphs

For a simple graph $G$, let $\chi_f(G)$ be the fractional chromatic number of $G$. In this paper, we aim to establish upper bounds on $\chi_f(G)$ for those graphs $G$ with restrictions on the clique number. Namely, we prove that for $\Delta \geq 4$, if $G$ has maximum degree at most $\Delta$ and is $K_{\Delta}$-free, then $\chi_f(G) \leq \Delta-\tfrac{1}{8}$ unless $G= C^2_8$ or $G = C_5\boxtimes K_2$. This im proves the result in [King, Lu, and Peng, SIAM J. Discrete Math., 26(2) (2012), pp. 452-471] for $\Delta \geq 4$ and the result in [Katherine and King, SIAM J.Discrete Math., 27(2) (2013), pp. 1184-1208] for $\Delta \in \{6,7,8\}$.