Improved bounds for zeros of the chromatic polynomial on bounded degree graphs

Abstract. We prove that for any graph G of maximum degree at most ∆, the zeros of its chromatic polynomial χG(z) (in C) lie outside the disk of radius 5.02∆ centered at 0. This improves on the previously best known bound of approximately 6.91∆. In the case of graphs of high girth we can improve this. We prove that for every g there is a constant Kg such that for any graph G of maximum degree at most ∆ and girth at least g, the zeros of its chromatic polynomial χG(z) lie outside the disk of radius Kg∆ centered at 0 where Kg → 1 + e ≈ 3.72 as g → ∞. Finally, we give improved bounds on the Fisher zeros of the partition function of the Ising model.