Bounding the fractional chromatic number of $K_\Delta$-free graphs

TL;DR

This paper improves bounds on the fractional chromatic number of $K_ riangle$-free graphs with maximum degree $ riangle$, using a weighted local approach related to Reed's conjecture, and discusses related conjectures.

Contribution

It provides improved bounds for $ riangle extgreater= 6$ and introduces a weighted local generalization of Reed's conjecture for fractional chromatic numbers.

Findings

Improved bounds for fractional chromatic number when $ riangle extgreater= 6$.

A new weighted local approach based on Reed's conjecture.

Identification of specific graphs where bounds are tight.

Abstract

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.