Chromatic roots at 2 and at the Beraha number B₁₀

2018/07/23 by Harvey, Daniel J., Royle, Gordon F.
#05C31 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1807.08397

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 B₁₀ = (5 + √(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].

Related

• The Beraha number B₁₀ is a chromatic root
• Graphs with chromatic roots in the interval (1,2)
• Polynomial bounds for chromatic number. V. Excluding a tree of radius two and a complete multipartite graph
• A set of chromatic roots which is dense in the complex plane and closed\n under multiplication by positive integers
• Some applications of the chromatic polynomials