A coloring of the vertices of a graph G is nonrepetitive if no path in G forms a sequence consisting of two identical blocks. The minimum number of colors needed is the Thue chromatic number, denoted by π(G). A famous theorem of Thue asserts that π(P) = 3 for any path P with at least 4 vertices. In this paper we study the Thue chromatic number of trees. In view of the fact that π(T) is bounded by 4 in this class we aim to describe the 4-chromatic trees. In particular, we study the 4-critical trees which are minimal with respect to this property. Though there are many trees T with π(T) = 4 we show that any of them has a sufficiently large subdivision H such that π(H) = 3. The proof relies on Thue sequences with additional properties involving palindromic words. We also investigate nonrepetitive edge colorings of trees. By a similar argument we prove that any tree has a subdivision which can be edge-colored by at most ∆ + 1 colors without repetitions on paths.

Many of Axel Thue's ideas have been influential in theoretical computer science. In particular, Thue systems, semi-Thue systems and his work on the combinatorics of words are well-known. Here we consider his 1910 paper which contains many notions and ideas about trees, term rewriting and word problems which are surprisingly modern and have later come to play important roles in mathematics, logic, and computer science.

We study a sequence, c, which encodes the lengths of blocks in the Thue-Morse sequence. In particular, we show that the generating function for c is a simple product. Consider the sequence

Abstract. We construct infinite cubefree binary words containing exponentially many distinct squares of length n. We also show that for every positive integer n, there is a cubefree binary square of length 2n. 1.