A Fractional Analogue of Brooks' Theorem

Let ∆(G) be the maximum degree of a graph G. Brooks’ theorem states that the only connected graphs with chromatic number χ(G) = ∆(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 χf (G) is at least ∆(G). These graphs are complete graphs, odd cycles, C 2 8 , C5 ⊠ K2, and graphs whose clique number ω(G) equals the maximum degree ∆(G). Among the two sporadic graphs, the graph C 8 is the square graph of cycle C8 while the other graph C5 ⊠ K2 is the strong product of C5 and K2. In fact, we prove a stronger result; if a connected graph G with ∆(G) ≥ 3 is not one of the graphs listed above, then we have χf (G) ≤ ∆(G)− 1 4(∆(G)−1)5 .