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Summary: NON-REPETITIVE COLORINGS OF GRAPHS
NOGA ALON, JAROSLAW GRYTCZUK, MARIUSZ HALUSZCZAK, AND OLIVER
RIORDAN
Abstract. A sequence a = a1a2...an is said to be non-repetitive if no two
adjacent blocks of a are exactly the same. For instance the sequence 1232321
contains a repetition 2323, while 123132123213 is non-repetitive. A theorem of
Thue asserts that, using only three symbols, one can produce arbitrarily long
non-repetitive 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 non-repetitive if the sequence of colors on any path in
G is non-repetitive. We call the minimal number of colors needed for such a
coloring the Thue number of G and denote it by (G).
The main problem we consider is the relation between the numbers (G)
and (G). We show, by an application of the Lov´asz Local Lemma, that the
Thue number stays bounded for graphs with bounded maximum degree, in
particular, (G) c(G)2 for some absolute constant c. For certain special
classes of graphs we obtain linear upper bounds on (G), by giving explicit
colorings. For instance, the Thue number of the complete graph Kn is at most
2n - 3, and (T) 4((T) - 1) for any tree T with at least two edges. We
conclude by discussing some generalizations and proposing several problems
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