A Fractional Analogue of Brooks' Theorem

TL;DR

This paper establishes a fractional analogue of Brooks' theorem, classifying connected graphs with fractional chromatic number at least their maximum degree, and providing bounds for other graphs.

Contribution

It introduces a fractional version of Brooks' theorem, identifying specific graphs with high fractional chromatic number and bounding the fractional chromatic number for others.

Findings

Classified all connected graphs with fractional chromatic number at least their maximum degree.

Identified specific sporadic graphs: $C^2_8$ and $C_5\boxtimes K_2$.

Proved a bound of $\chi_f(G) \leq \Delta(G) - 2/67$ for non-listed graphs with $\Delta(G) \geq 4$.

Abstract

Let $\Delta(G)$ be the maximum degree of a graph $G$. Brooks' theorem states that the only connected graphs with chromatic number $\chi(G)=\Delta(G)+1$ are complete graphs and odd cycles. We prove a fractional analogue of Brooks' theorem in this paper. Namely, we classify all connected graphs $G$ such that the fractional chromatic number $\chi_f(G)$ is at least $\Delta(G)$. These graphs are complete graphs, odd cycles, $C^2_8$, $C_5\boxtimes K_2$, and graphs whose clique number $\omega(G)$ equals the maximum degree $\Delta(G)$. Among the two sporadic graphs, the graph $C^2_8$ is the square graph of cycle $C_8$ while the other graph $C_5\boxtimes K_2$ is the strong product of $C_5$ and $K_2$. In fact, we prove a stronger result; if a connected graph $G$ with $\Delta(G)\geq 4$ is not one of the graphs listed above, then we have $\chi_f(G)\leq \Delta(G)- 2/67$.