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Chromatic roots as algebraic integers

Abstract : A chromatic root is a zero of the chromatic polynomial of a graph. At a Newton Institute workshop on Combinatorics and Statistical Mechanics in 2008, two conjectures were proposed on the subject of which algebraic integers can be chromatic roots, known as the ``$α +n$ conjecture'' and the ``$nα$ conjecture''. These say, respectively, that given any algebraic integer α there is a natural number $n$ such that $α +n$ is a chromatic root, and that any positive integer multiple of a chromatic root is also a chromatic root. By computing the chromatic polynomials of two large families of graphs, we prove the $α +n$ conjecture for quadratic and cubic integers, and show that the set of chromatic roots satisfying the nα conjecture is dense in the complex plane.
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Adam Bohn. Chromatic roots as algebraic integers. 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), 2012, Nagoya, Japan. pp.539-550, ⟨10.46298/dmtcs.3061⟩. ⟨hal-01283106⟩



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