Chromatic roots are dense in the whole complex plane

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.Comment: 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 several small improvements. To appear in Combinatorics, Probability & Computin

Chromatic roots are dense in the whole complex plane

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 - l\ < l. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Potts-model partition functions) Z(G)(q,upsilon) outside the disc \q + upsilon\ < \upsilon\. An immediate corollary is that the chromatic roots 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 oil the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof

To Prove Four Color Theorem

In this paper, we give a proof for four color theorem(four color conjecture). Our proof does not involve computer assistance and the most important is that it can be generalized to prove Hadwiger Conjecture. Moreover, we give algorithms to color and test planarity of planar graphs, which can be generalized to graphs containing $K_x(x>5)$ minor. There are four parts of this paper: Part-1: To Prove Four Color Theorem Part-2: An Equivalent Statement of Hadwiger Conjecture when $k=5$ Part-3: A New Proof of Wagner's Equivalence Theorem Part-4: A Geometric View of Outerplanar GraphComment: The paper is further reduced, and each part is more self-contained, is the fina