# Two conjectures on chromatic roots

Speaker:
A zero of the chromatic polynomial of a graph is known as a chromatic root. These have been the subject of much study by combinatorialists and analysts; in particular, a lot is known about how chromatic roots are distributed in the complex plane and real line. However, the question of which algebraic integers can be chromatic roots is still very much open. At the Newton Institute in 2008, Peter Cameron proposed two conjectures on this subject. These say, firstly, that for any algebraic integer $\alpha$ there is a natural number $n$ such that $\alpha +n$ is a chromatic root, and secondly, that any positive integer multiple of a chromatic root is also a chromatic root. I will briefly summarise the background to these problems, and present some recent progress I have made on them.