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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.
There are no iterated morphisms that define the Arshon sequence and the sigmasequence, to appear
 J. Automata, Languages and Combinatorics
, 2002
"... sequence and the σsequence ..."
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.
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.
Pseudopower Avoidance
, 2012
"... Repetition avoidance has been intensely studied since Thue’s work in the early 1900’s. In this paper, we consider another type of repetition, called pseudopower, inspired by the WatsonCrick complementarity property of DNA sequences. A DNA single strand can be viewed as a string over the fourletter ..."
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Repetition avoidance has been intensely studied since Thue’s work in the early 1900’s. In this paper, we consider another type of repetition, called pseudopower, inspired by the WatsonCrick complementarity property of DNA sequences. A DNA single strand can be viewed as a string over the fourletter alphabet{A,C,G,T}, wherein A is the complement of T, while C is the complement of G. Such a DNA single strand will bind to a reverse complement DNA single strand, called its WatsonCrick complement, to form a helical doublestranded DNA molecule. The WatsonCrick complement of a DNA strand is deducible from, and thus informationally equivalent to, the original strand. We use this fact to generalize the notion of the power of a word by relaxing the meaning of “sameness ” to include the image through an antimorphic involution, the model of DNA WatsonCrick complementarity. Given a finite alphabet Σ, an antimorphic involution is a function θ: Σ ∗ −→ Σ ∗ which is an involution, i.e.,θ 2 equals the identity, and an antimorphism, i.e., θ(uv) = θ(v)θ(u), for all u ∈ Σ ∗. For a positive integer k, we call a word w a pseudokthpower with respect to θ if it can be written as w = u1...uk, where for 1 ≤ i,j ≤ k we have either ui = uj or ui = θ(uj). The classical kthpower of a word is a special case of a pseudokthpower, where all the repeating units are identical. We first classify the alphabets Σ and the antimorphic involutions θ for which
Words avoiding abelian inclusions
 J. Automata, Languages and Combinatorics. OttovonGuerickeUniv
"... ABSTRACT We study a generalization of abelian squares which we call abelian inclusions: a word uv is said to be an f(l)inclusion if the commutative image of v majorizes that of u, and jvj ^ juj + f(juj). We prove that clinclusions are unavoidable, but cinclusions are avoidable for an arbitrary co ..."
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ABSTRACT We study a generalization of abelian squares which we call abelian inclusions: a word uv is said to be an f(l)inclusion if the commutative image of v majorizes that of u, and jvj ^ juj + f(juj). We prove that clinclusions are unavoidable, but cinclusions are avoidable for an arbitrary constant c.
Improved Bounds on the Length of Maximal Abelian SquareFree Words
, 2004
"... A word is abelian squarefree if it does not contain two adjacent subwords which are permutations of each other. Over an alphabet k on k letters, an abelian squarefree word is maximal if it cannot be extended to the left or right by letters from k and remain abelian squarefree. Michael Korn pr ..."
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A word is abelian squarefree if it does not contain two adjacent subwords which are permutations of each other. Over an alphabet k on k letters, an abelian squarefree word is maximal if it cannot be extended to the left or right by letters from k and remain abelian squarefree. Michael Korn proved that the length `(k) of a shortest maximal abelian squarefree word satis es 4k 7 `(k) 6k 10 for k 6. In this paper, we re ne Korn's methods to show that 6k 29 `(k) 6k 12 for k 8.
unknown title
, 2002
"... There are no iterated morphisms that define the Arshon sequence and the oesequence ..."
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There are no iterated morphisms that define the Arshon sequence and the oesequence