On Brooks' Theorem for Sparse Graphs

Let G be a graph with maximum degree ∆(G). In this paper we prove that if the girth g(G) of G is greater than 4 then its chromatic number, χ(G), satisfies χ(G) ≤ (1 + o(1)) ∆(G) log ∆(G) where o(1) goes to zero as ∆(G) goes to infinity. (Our logarithms are base e.) More generally, we prove the same bound for the list-chromatic (or choice) number: χ l (G) ≤ (1 + o(1)) ∆(G) log ∆(G) provided g(G) > 4.