Abstract not found

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 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.

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 1-2, 2-1, 1-1- · · ·-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 1-2, 2-1, 12, 21, 123 and 1-1- · · ·-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.

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.

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 cl-inclusions are unavoidable, but c-inclusions are avoidable for an arbitrary constant c.

A word is abelian square-free 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 square-free. Michael Korn proved that the length `(k) of a shortest maximal abelian square-free 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.

There are no iterated morphisms that define the Arshon sequence and the oe-sequence

Abstract. A word is crucial with respect to a given set of prohibited words (or simply prohibitions) if it avoids the prohibitions but it cannot be extended to the right by any letter of its alphabet without creating a prohibition. A minimal crucial word is a crucial word of the shortest length. A word W contains an abelian k-th power if W has a factor of the form X1X2... Xk where Xi is a permutation of X1 for 2 ≤ i ≤ k. When k = 2 or 3, one deals with abelian squares and abelian cubes, respectively. Evdokimov and Kitaev [6] have shown that a minimal crucial word over an n-letter alphabet An = {1, 2,..., n} avoiding abelian squares has length 4n − 7 for n ≥ 3. In this paper, we show that a minimal crucial word over An avoiding abelian cubes has length 9n−13 for n ≥ 5, and it has length 2, 5, 11, and 20 for n = 1,2, 3, and 4, respectively. Moreover, for n ≥ 4 and k ≥ 2, we give a construction of length k 2 (n−1)−k −1 of a crucial word over An avoiding abelian k-th powers. This construction gives the minimal length for k = 2 and k = 3. For k ≥ 4 and n ≥ 5, we provide a lower bound for the length of crucial words over An avoiding abelian k-th powers. 1.

Abstract not found

are exponentially many ternary words that avoid