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Bounds For The Real Zeros of Chromatic Polynomials

Sokal in 2001 proved that the complex zeros of the chromatic polynomial PG(q) of any graph G lie in the disc |q| < 7.963907Δ, where Δ is the maximum degree of G. This result answered a question posed by Brenti, Royle and Wagner in 1994 and hence proved a conjecture proposed by Biggs, Damerell and Sands in 1972. Borgs gave a short proof of Sokal's result. Fernández and Procacci recently improved Sokal's result to |q| < 6.91Δ. In this paper, we shall show that all real zeros of PG(q) are in the interval [0,5.664Δ). For the special case that Δ = 3, all real zeros of PG(q) are in the interval [0,4.765Δ).