 Alon et al.use the Lovász local lemma to prove that the Thue number of any graph is at most quadratic in its maximum degree; they provide an example showing that for some graphs this quadratic dependence is necessary.In addition they show that the Thue number of a path of four or more vertices is exactly three, and that the Thue number of any cycle is at most four, and that the Thue number of the Petersen graph is exactly five.