Chromatic roots at 2 and at the Beraha number $B_{10}$

Abstract

By the construction of suitable graphs and the determination of their chromatic polynomials, we resolve two open questions concerning real chromatic roots. First we exhibit graphs for which the Beraha number $B_{10} = (5 + \sqrt{5})/2$ is a chromatic root. As it was previously known that no other non-integer Beraha number is a chromatic root, this completes the determination of precisely which Beraha numbers can be chromatic roots. Next we construct an infinite family of $3$-connected graphs such that for any $k \geqslant 1$, there is a member of the family with $q=2$ as a chromatic root of multiplicity at least $k$. The former resolves a question of Salas and Sokal [J. Statist. Pys. 104 (2001) pp. 609--699] and the latter a question of Dong and Koh [J. Graph Theory 70 (2012) pp. 262--283].