The Beraha number $B_{10}$ is a chromatic root

TL;DR

This paper demonstrates graphs with chromatic polynomials that have the Beraha number B_{10} as a root, completing the classification of Beraha numbers that can be chromatic roots.

Contribution

It provides explicit examples of graphs with B_{10} as a chromatic root, resolving the question of which non-integer Beraha numbers are chromatic roots.

Findings

Graphs with B_{10} as a chromatic root are constructed.

No other non-integer Beraha numbers are chromatic roots.

The classification of Beraha numbers as chromatic roots is complete.

Abstract

This note exhibits graphs whose chromatic polynomials have the Beraha number $B_{10} = (5 + \sqrt{5})/2$ as a root. It was previously known that no other non-integer Beraha number is a chromatic root, and so these examples complete the determination of precisely which Beraha numbers can be chromatic roots.