On chromatic polynomials and the golden ratio

Abstract

Let M be a triangulation of the 2-sphere, with k vertices. Let P(M, n) be its chromatic polynomial with respect to vertex-colorings. Then| P (M, 1 + τ) | ≤ τ5 - kwhere τ is the "golden ratio" (1 + sqrt(5)) / 2. This result is offered as a theoretical explanation of the empirical observation that P(M, n) tends to have a zero near n=1+τ (see [1]). © 1970 Academic Press, Inc.