### Abstract

I show that the zeros of the chromatic polynomials P
_{G}(q) for the generalized theta graphs Θ
^{(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 |3q + v| < |v|. 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 on the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof.

Original language | English (US) |
---|---|

Pages (from-to) | 221-261 |

Number of pages | 41 |

Journal | Combinatorics Probability and Computing |

Volume | 13 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2004 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Computational Theory and Mathematics
- Mathematics(all)
- Discrete Mathematics and Combinatorics
- Statistics and Probability

### Cite this

*Combinatorics Probability and Computing*,

*13*(2), 221-261. https://doi.org/10.1017/S0963548303006023

**Chromatic roots are dense in the whole complex plane.** / Sokal, Alan D.

Research output: Contribution to journal › Article

*Combinatorics Probability and Computing*, vol. 13, no. 2, pp. 221-261. https://doi.org/10.1017/S0963548303006023

}

TY - JOUR

T1 - Chromatic roots are dense in the whole complex plane

AU - Sokal, Alan D.

PY - 2004/3

Y1 - 2004/3

N2 - I show that the zeros of the chromatic polynomials P G(q) for the generalized theta graphs Θ (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 |3q + v| < |v|. 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 on the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof.

AB - I show that the zeros of the chromatic polynomials P G(q) for the generalized theta graphs Θ (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 |3q + v| < |v|. 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 on the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof.

UR - http://www.scopus.com/inward/record.url?scp=3142645491&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=3142645491&partnerID=8YFLogxK

U2 - 10.1017/S0963548303006023

DO - 10.1017/S0963548303006023

M3 - Article

AN - SCOPUS:3142645491

VL - 13

SP - 221

EP - 261

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

SN - 0963-5483

IS - 2

ER -