### 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) |
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Pages (from-to) | 221-261 |

Number of pages | 41 |

Journal | Combinatorics Probability and Computing |

Volume | 13 |

Issue number | 2 |

DOIs | |

State | Published - Mar 1 2004 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics

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## Cite this

*Combinatorics Probability and Computing*,

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