A Fractional Analogue of Brooks' Theorem
Andrew D. King, Linyuan Lü, Xing Peng · SIAM Journal on Discrete Mathematics · 2012
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 $\chi_f(G)\leq \Delta(G)- \frac{2}{67}$.