Chromatic roots are dense in the whole complex plane

Details Chromatic roots are dense in the whole complex plane Chromatic roots are dense in the whole complex plane

2000-12-19

arxiv.org/abs/cond-mat/0012369v3

Combin. Probab. Comput. 13, 221-261 (2004)

Physics

Condensed Matter

Statistical Mechanics

LaTeX2e, 53 pages. Version 2 includes a new Appendix B. Version 3 adds a new Theorem 1.4 and a new Section 5, and makes severa

Scientific paper

I show that the zeros of the chromatic polynomials P_G(q) for the generalized theta graphs \Theta^{(s,p)} are, taken together, dense in the whole complex plane with the possible exception of the disc |q-1| < 1. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Potts-model partition functions) Z_G(q,v) outside the disc |q+v| < |v|. An immediate corollary is that the chromatic zeros of not-necessarily-planar graphs are dense in the whole complex plane. The main technical tool in the proof of these results is the Beraha-Kahane-Weiss theorem on the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof.

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