Improved bounds for the zeros of the chromatic polynomial via Whitney's Broken Circuit Theorem
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We prove that for any graph G of maximum degree at most Δ, the zeros of its chromatic polynomial χ_G(x) (in ℂ) lie inside the disc of radius 5.94 Δ centered at 0. This improves on the previously best known bound of approximately 6.91Δ. We also obtain improved bounds for graphs of high girth. We prove that for every g there is a constant K_g such that for any graph G of maximum degree at most Δ and girth at least g, the zeros of its chromatic polynomial χ_G(x) lie inside the disc of radius K_g Δ centered at 0, where K_g is the solution to a certain optimization problem. In particular, K_g < 5 when g ≥ 5 and K_g < 4 when g ≥ 25 and K_g tends to approximately 3.86 as g →∞. Key to the proof is a classical theorem of Whitney which allows us to relate the chromatic polynomial of a graph G to the generating function of so-called broken-circuit-free forests in G. We also establish a zero-free disc for the generating function of all forests in G (aka the partition function of the arboreal gas) which may be of independent interest.
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