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The origins of combinatorics on words
, 2007
"... We investigate the historical roots of the field of combinatorics on words. They comprise applications and interpretations in algebra, geometry and combinatorial enumeration. These considerations gave rise to early results such as those of Axel Thue at the beginning of the 20th century. Other early ..."
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We investigate the historical roots of the field of combinatorics on words. They comprise applications and interpretations in algebra, geometry and combinatorial enumeration. These considerations gave rise to early results such as those of Axel Thue at the beginning of the 20th century. Other early results were obtained as a byproduct of investigations on various combinatorial objects. For example, paths in graphs are encoded by words in a natural way, and conversely, the Cayley graph of a group or a semigroup encodes words by paths. We give in this text an account of this twosided interaction.
NONREPETITIVE COLORINGS OF TREES
"... 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 ..."
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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 4chromatic trees. In particular, we study the 4critical 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 edgecolored by at most ∆ + 1 colors without repetitions on paths.
Sequence related to ThueMorse Send proofs to:
, 1997
"... We study a sequence, c, which encodes the lengths of blocks in the ThueMorse sequence. In particular, we show that the generating function for c is a simple product. Consider the sequence ..."
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We study a sequence, c, which encodes the lengths of blocks in the ThueMorse sequence. In particular, we show that the generating function for c is a simple product. Consider the sequence
CUBEFREE WORDS WITH MANY SQUARES
, 811
"... 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. ..."
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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.
for kpowerfreeness of uniform morphisms
, 2008
"... A challenging problem is to find an algorithm to decide whether a morphism is kpowerfree. We provide such an algorithm when k ≥ 3 for uniform morphisms showing that in such a case, contrarily to the general case, there exist finite testsets for kpowerfreeness. ..."
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A challenging problem is to find an algorithm to decide whether a morphism is kpowerfree. We provide such an algorithm when k ≥ 3 for uniform morphisms showing that in such a case, contrarily to the general case, there exist finite testsets for kpowerfreeness.
Nonrepetitive sequences on arithmetic progressions
"... A sequence S = s1s2...sn is said to be nonrepetitive if no two adjacent blocks of S are identical. In 1906 Thue proved that there exist arbitrarily long nonrepetitive sequences over 3element set of symbols. We study a generalization of nonrepetitive sequences involving arithmetic progressions. We p ..."
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A sequence S = s1s2...sn is said to be nonrepetitive if no two adjacent blocks of S are identical. In 1906 Thue proved that there exist arbitrarily long nonrepetitive sequences over 3element set of symbols. We study a generalization of nonrepetitive sequences involving arithmetic progressions. We prove that for every k � 1, there exist arbitrarily long sequences over at most 2k +10 √ k symbols whose subsequences, indexed by arithmetic progressions with common differences from the set {1,2,...,k}, are nonrepetitive. This improves a previous bound of e 33 k obtained by