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Weak Repetitions In Strings
 J. Combinatorial Mathematics and Combinatorial Computing
"... A weak repetition in a string consists of two or more adjacent substrings which are permutations of each other. We describe a straightforward \Theta(n 2 ) algorithm which computes all the weak repetitions in a given string of length n defined on an arbitrary alphabet A. Using results on Fibonacci ..."
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A weak repetition in a string consists of two or more adjacent substrings which are permutations of each other. We describe a straightforward \Theta(n 2 ) algorithm which computes all the weak repetitions in a given string of length n defined on an arbitrary alphabet A. Using results on Fibonacci and other simple strings, we prove that this algorithm is asymptotically optimal over all known encodings of the output. 1 INTRODUCTION Interest in the periodic behaviour of strings dates back to Thue [T06] at the turn of the century. Thue considered what we call here strong repetitions (equal adjacent substrings) and showed how to construct an infinitely long string on an alphabet of only three letters with no strong repetitions. (Other constructions on three letters have been discovered several times since, most recently by Dekking [D79] and Pleasants [P70]  the latter lists several references to earlier constructions.) More recently, Erdos [E61, p. 240] considered "Abelian squares" (w...
NONREPETITIVE COLORINGS OF GRAPHS
"... A sequence a = a1a2...an is said to be nonrepetitive if no two adjacent blocks of a are exactly the same. For instance the sequence 1232321 contains a repetition 2323, while 123132123213 is nonrepetitive. A theorem of Thue asserts that, using only three symbols, one can produce arbitrarily long no ..."
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A sequence a = a1a2...an is said to be nonrepetitive if no two adjacent blocks of a are exactly the same. For instance the sequence 1232321 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 ∆(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∆(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 Kn is at most 2n − 3, and π(T) ≤ 4(∆(T) − 1) for any tree T with at least two edges. We conclude by discussing some generalizations and proposing several problems and conjectures.
On Uniform DOL Words
 STACS'98, LNCS 1373
, 1998
"... . We introduce the wide class of marked uniform DOL words and study their structure. The criterium of circularity of a marked uniform DOL word is given, and the subword complexity function is found for the uncircular case as well as for the circular one. The same technique is valid for a wider c ..."
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. We introduce the wide class of marked uniform DOL words and study their structure. The criterium of circularity of a marked uniform DOL word is given, and the subword complexity function is found for the uncircular case as well as for the circular one. The same technique is valid for a wider class of uniform DOL sequences which includes (p; 1)Toeplitz words (see [4]). 1 Introduction DOL word w is an infinite word on a finite alphabet \Sigma which is a fixed point of a morphism ' : \Sigma ! \Sigma ; i. e., w = lim i!1 ' i (a) for a 2 \Sigma . The class of DOL words has been extensively studied and contains famous examples concerning pattern avoidance, such as the cubefree ThueMorse word on the twoletter alphabet and a squarefree word on the threeletter alphabet. A DOL word w is called uniform if all the words '(a i ); a i 2 \Sigma , are of the same length. In this paper, we deal with marked DOL words; i. e., fixed points of uniform morphisms with all the images of...
Repetitive Perhaps, But Certainly Not Boring
"... In this paper some of the work done on repetitions in strings is surveyed, especially that of an algorithmic nature. Several open problems are described and conjectures formulated about some of them. KEYWORDS: string, word, repetition, repeat, cover 1 INTRODUCTION Repetitions in strings are usual ..."
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In this paper some of the work done on repetitions in strings is surveyed, especially that of an algorithmic nature. Several open problems are described and conjectures formulated about some of them. KEYWORDS: string, word, repetition, repeat, cover 1 INTRODUCTION Repetitions in strings are usually thought of as adjacent or \tandem"; that is, the string uvu is counted as a repetition of u if and only if v = , the empty string. However, in certain contexts  for example, DNA sequence analysis [S98], data compression [IS98], analysis of musical texts [CIR96]  this denition may be too narrow. Here therefore we take a wider view and regard uvu as a repetition of a nonempty string u for any nite string v. Even more generally, we also count as repetitions cases where the string u overlaps itself; for example, abaabaab is accepted as a repetition of abaab. In order to make sure that these ideas are clear, we express them more formally. Throughout this paper x will denote a string of l...
A survey on certain pattern problems
"... Abstract. The paper contains all the definitions and notations needed to understand the results concerning the field dealing with occurrences of patterns in permutations and words. Also, this paper includes a historical overview on the results obtained in this subject. The authors tried to collect a ..."
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Abstract. The paper contains all the definitions and notations needed to understand the results concerning the field dealing with occurrences of patterns in permutations and words. Also, this paper includes a historical overview on the results obtained in this subject. The authors tried to collect all the currently existing references to the papers directly related to the subject. Moreover, a number of basic approaches to study the pattern problems are discussed.
Counting the occurrences of generalized patterns in words generated by a morphism
, 2002
"... We count the number of occurrences of certain patterns in given words. We choose these words to be the set of all finite approximations of a sequence generated by a morphism with certain restrictions. The patterns in our considerations are either classical patterns 12, 21, 11 · · ·1, or arbit ..."
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We count the number of occurrences of certain patterns in given words. We choose these words to be the set of all finite approximations of a sequence generated by a morphism with certain restrictions. The patterns in our considerations are either classical patterns 12, 21, 11 · · ·1, or arbitrary generalized patterns without internal dashes, in which repetitions of letters are allowed. In particular, we find the number of occurrences of the patterns 12, 21, 12, 21, 123 and 11 · · ·1 in the words obtained by iterations of the morphism 1 → 123, 2 → 13, 3 → 2, which is a classical example of a morphism generating a nonrepetitive sequence.
SPLITTING NECKLACES AND MEASURABLE COLORINGS OF THE REAL LINE
"... Abstract. A (continuous) necklace is simply an interval of the real line colored measurably with some number of colors. A wellknown application of the BorsukUlam theorem asserts that every kcolored necklace can be fairly split by at most k cuts (from the resulting pieces one can form two collecti ..."
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Abstract. A (continuous) necklace is simply an interval of the real line colored measurably with some number of colors. A wellknown application of the BorsukUlam theorem asserts that every kcolored necklace can be fairly split by at most k cuts (from the resulting pieces one can form two collections, each capturing the same measure of every color). Here we prove that for every k ≥ 1thereisameasurable(k+3)coloring of the real line such that no interval can be fairly split using at most k cuts. In particular, there is a measurable 4coloring of the real line in which no two adjacent intervals have the same measure of every color. An analogous problem for the integers was posed by Erdős in 1961 and solved in the affirmative by Keränen in 1991. Curiously, in the discrete case the desired coloring also uses four colors. 1.
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