Nonrepetitive colorings of graphs

A sequence a = a 1 a 2 … a n is said to be nonrepetitive if no two adjacent blocks of a are exactly the same. For instance, the sequence 1 2323 21 contains a repetition 2323, while 123132123213 is nonrepetitive. A theorem of Thue asserts that, using only three symbols, one can produce arbitrarily long nonrepetitive sequences. In this paper we consider a natural generalization of Thue's sequences for colorings of graphs. A coloring of the set of edges of a given graph G is nonrepetitive if the sequence of colors on any path in G is nonrepetitive. We call the minimal number of colors needed for such a coloring the Thue number of G and denote it by π( G ). The main problem we consider is the relation between the numbers π( G ) and Δ( G ). We show, by an application of the Lovász Local Lemma, that the Thue number stays bounded for graphs with bounded maximum degree, in particular, π( G ) ≤ c Δ( G ) 2 for some absolute constant c. For certain special classes of graphs we obtain linear upper bounds on π( G ), by giving explicit colorings. For instance, the Thue number of the complete graph K n is at most 2 n − 3, and π( T ) ≤ 4(Δ( T ) − 1)for any tree T with at least two edges. We conclude by discussing some generalizations and proposing several problems and conjectures. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 336–346, 2002