Bounding the Fractional Chromatic Number of $K_\Delta$-Free Graphs
Katherine Edwards, Andrew D. King · SIAM Journal on Discrete Mathematics · 2013
King, Lu, and Peng recently proved that for $\Delta\geq 4$, any $K_\Delta$-free graph with maximum degree $\Delta$ has fractional chromatic number at most $\Delta-\tfrac{2}{67}$ unless it is isomorphic to $C_5\boxtimes K_2$ or $C_8^2$. Using a different approach we give improved bounds for $\Delta\geq 6$ and pose several related conjectures. Our proof relies on a weighted local generalization of the fractional relaxation of Reed's $\omega$, $\Delta$, $\chi$ conjecture.