A Fractional Analogue of Brooks' Theorem

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, $C_8^2$, $C_5⊠K_2$, and graphs whose clique number $\omega (G)$ equals the maximum degree $\Delta (G)$. Among the two sporadic graphs, the graph $C_8^2$ is the square graph of cycle $C_8$ while the other graph $C_5⊠K_2$ is the strong product of $C_5$ and $K_2$. In fact, we prove a stronger result; if a connected graph $G$ with $\Delta (G)\ge 4$ is not one of the graphs listed above, then we have ${\chi }_f(G)\le \Delta (G)-2/67$.

arXiv commentThird version, add Andrew King as an coauthor