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Axel Thue's work on repetitions in words
 Invited Lecture at the 4th Conference on Formal Power Series and Algebraic Combinatorics
, 1992
"... The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched. ..."
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The purpose of this survey is to present, in contemporary terminology, the fundamental contributions of Axel Thue to the study of combinatorial properties of sequences of symbols, insofar as repetitions are concerned. The present state of the art is also sketched.
Extremal Infinite OverlapFree Binary Words
 Electronic J. Combinatorics
, 1997
"... Let t be the infinite fixed point, starting with 1, of the morphism :0#01, 1 # 10. An infinite word over {0, 1} is said to be overlapfree if it contains no factor of the form axaxa, where a #{0,1}and x #{0,1} # . We prove that the lexicographically least infinite overlapfree binary word begi ..."
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Let t be the infinite fixed point, starting with 1, of the morphism :0#01, 1 # 10. An infinite word over {0, 1} is said to be overlapfree if it contains no factor of the form axaxa, where a #{0,1}and x #{0,1} # . We prove that the lexicographically least infinite overlapfree binary word beginning with any specified prefix, if it exists, has a su#x which is a su#x of t. In particular, the lexicographically least infinite overlapfree binary word is 001001t. Keywords: Homomorphism, fixed point, overlapfree word. 1991 Mathematics Subject Classification: Primary 68R15. 1 the electronic journal of combinatorics 5 (1998),#R27 2 1 Introduction Since the pioneering work of Thue [14, 15] (see also [5]) the overlapfree words on a finite alphabet, i.e., those words that do not contain a factor axaxa,wherexis a word and a a letter, have been studied extensively. The question of extremality (for the lexicographic order) of overlapfree binary infinite words seems to have been addre...
Nonrepetitive Colorings of Graphs  A Survey
, 2007
"... A vertex coloring f of a graph G is nonrepetitive if there are no integer r ≥ 1 and a simple path v1,...,v2r in G such that f (vi) = f (vr+i) for all i = 1,...,r. This notion is a graphtheoretic variant of nonrepetitive sequences of Thue. The paper surveys problems and results on this topic. ..."
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A vertex coloring f of a graph G is nonrepetitive if there are no integer r ≥ 1 and a simple path v1,...,v2r in G such that f (vi) = f (vr+i) for all i = 1,...,r. This notion is a graphtheoretic variant of nonrepetitive sequences of Thue. The paper surveys problems and results on this topic.
Words strongly avoiding fractional powers
 Europ. J. of Combinatorics
, 1999
"... Abstract Let k be fixed, 1! k! 2. There exists an infinite word over a finite alphabet which contains no subword of the form xyz with jxyzj=jxyj * k and where z is a permutation of x. 1 Introduction Nonrepetitive words have been studied since Thue [34]. A word is nonrepetitive if it cannot be writt ..."
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Abstract Let k be fixed, 1! k! 2. There exists an infinite word over a finite alphabet which contains no subword of the form xyz with jxyzj=jxyj * k and where z is a permutation of x. 1 Introduction Nonrepetitive words have been studied since Thue [34]. A word is nonrepetitive if it cannot be written xyyz, where x; y; z are words, and y is nonempty. Infinite
A Note on Antichains of Words
, 1995
"... We can compress the word `banana' as xyyz, where x = `b', y = `an',z = `a'. We say that `banana' encounters yy. Thus a `coded' version of yy shows up in `banana'. The relation `u encounters w' is transitive, and thus generates an order on words. We study an ..."
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We can compress the word `banana' as xyyz, where x = `b', y = `an',z = `a'. We say that `banana' encounters yy. Thus a `coded' version of yy shows up in `banana'. The relation `u encounters w' is transitive, and thus generates an order on words. We study antichains under this order. In particular we show that in this order there is an infinite antichain of binary words avoiding overlaps. AMS Subject Classification: 68R15, 06A99 Key Words: overlaps. antichains, words avoiding patterns 1 Introduction The study of words avoiding patterns is an area of combinatorics on words reaching back at least to the turn of the century, when Thue proved [29] that one can find arbitrarily long words over a 3 letter alphabet in which no two adjacent subwords are identical. If w is such a word, then w cannot be written w = xyyz with y a nonempty word. In modern parlance, we would say that w avoids yy. A word which can be written as xyyz is said to encounter yy. Thue also showed that there are ar...