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

TL;DR

This paper establishes improved upper bounds on the fractional chromatic number of $K_{ riangle}$-free graphs with maximum degree at least 4, refining previous results and identifying specific exceptions.

Contribution

It proves tighter bounds on the fractional chromatic number for $K_{ riangle}$-free graphs with maximum degree at least 4, extending and improving earlier work.

Findings

For $ riangle eq 8$, $ eq 5$, the fractional chromatic number is at most $ riangle - rac{1}{8}$.

The bounds are tight except for specific graphs $C_8^2$ and $C_5 oxtimes K_2$.

Improves previous bounds for $ riangle eq 6,7,8$.

Abstract

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 $\Delta \in \{6,7,8\}$.