Bounding the fractional chromatic number of $K_Δ$-free graphs

Abstract

King, Lu, and Peng recently proved that for $Δ\geq 4$, any $K_Δ$-free graph with maximum degree $Δ$ has fractional chromatic number at most $Δ-\tfrac{2}{67}$ unless it is isomorphic to $C_5\boxtimes K_2$ or $C_8^2$. Using a different approach we give improved bounds for $Δ\geq 6$ and pose several related conjectures. Our proof relies on a weighted local generalization of the fractional relaxation of Reed's $ω$, $Δ$, $χ$ conjecture.

30 pages, revised edition