43 research outputs found

    Nonrepetitive colorings of lexicographic product of graphs

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    A coloring cc of the vertices of a graph GG is nonrepetitive if there exists no path v1v2…v2lv_1v_2\ldots v_{2l} for which c(vi)=c(vl+i)c(v_i)=c(v_{l+i}) for all 1≤i≤l1\le i\le l. Given graphs GG and HH with ∣V(H)∣=k|V(H)|=k, the lexicographic product G[H]G[H] is the graph obtained by substituting every vertex of GG by a copy of HH, and every edge of GG by a copy of Kk,kK_{k,k}. %Our main results are the following. We prove that for a sufficiently long path PP, a nonrepetitive coloring of P[Kk]P[K_k] needs at least 3k+⌊k/2⌋3k+\lfloor k/2\rfloor colors. If k>2k>2 then we need exactly 2k+12k+1 colors to nonrepetitively color P[Ek]P[E_k], where EkE_k is the empty graph on kk vertices. If we further require that every copy of EkE_k be rainbow-colored and the path PP is sufficiently long, then the smallest number of colors needed for P[Ek]P[E_k] is at least 3k+13k+1 and at most 3k+⌈k/2⌉3k+\lceil k/2\rceil. Finally, we define fractional nonrepetitive colorings of graphs and consider the connections between this notion and the above results

    Nonrepetitive colorings of lexicographic product of paths and other graphs

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    A coloring cc of the vertices of a graph GG is nonrepetitive if there exists no path v1v2…v2lv_1v_2\ldots v_{2l} for which c(vi)=c(vl+i)c(v_i)=c(v_{l+i}) for all 1≤i≤l1\le i\le l. Given graphs GG and HH with ∣V(H)∣=k|V(H)|=k, the lexicographic product G[H]G[H] is the graph obtained by substituting every vertex of GG by a copy of HH, and every edge of GG by a copy of Kk,kK_{k,k}. We prove that for a sufficiently long path PP, a nonrepetitive coloring of P[Kk]P[K_k] needs at least 3k+⌊k/2⌋3k+\lfloor k/2\rfloor colors. If k>2k>2 then we need exactly 2k+12k+1 colors to nonrepetitively color P[Ek]P[E_k], where EkE_k is the empty graph on kk vertices. If we further require that every copy of EkE_k be rainbow-colored and the path PP is sufficiently long, then the smallest number of colors needed for P[Ek]P[E_k] is at least 3k+13k+1 and at most 3k+⌈k/2⌉3k+\lceil k/2\rceil. Finally, we define fractional nonrepetitive colorings of graphs and consider the connections between this notion and the above results

    Dagstuhl Reports : Volume 1, Issue 2, February 2011

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    Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn

    Extensions and reductions of square-free words

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    A word is square-free if it does not contain a nonempty word of the form XXXX as a factor. A famous 1906 result of Thue asserts that there exist arbitrarily long square-free words over a 33-letter alphabet. We study square-free words with additional properties involving single-letter deletions and extensions of words. A square-free word is steady if it remains square-free after deletion of any single letter. We prove that there exist infinitely many steady words over a 44-letter alphabet. We also demonstrate that one may construct steady words of any length by picking letters from arbitrary alphabets of size 77 assigned to the positions of the constructed word. We conjecture that both bounds can be lowered to 44, which is best possible. In the opposite direction, we consider square-free words that remain square-free after insertion of a single (suitably chosen) letter at every possible position in the word. We call them bifurcate. We prove a somewhat surprising fact, that over a fixed alphabet with at least three letters, every steady word is bifurcate. We also consider families of bifurcate words possessing a natural tree structure. In particular, we prove that there exists an infinite tree of doubly infinite bifurcate words over alphabet of size 1212.Comment: 11 pages, 1 figur

    Répétitions dans les mots et seuils d'évitabilité

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    Nous étudions dans cette thèse différents problèmes d'évitabilité des répétitions dans les mots infinis. Soulevée par Thue et motivée par ses travaux sur les mots sans carrés, la problématique s'est développée au cours du XXe siècle, et est aujourd'hui devenue un des grands domaines de recherche en combinatoire des mots. En 1972, Dejean proposa une importante conjecture, dont la validation étape par étape s'est terminée récemment (2009). La conjecture concerne le seuil des répétitions d'un alphabet, i.e., la borne inférieure des exposants évitables sur cet alphabet. La notion de seuil, comme frontière entre évitabilité et non-évitabilité d'un ensemble donné de mots, est le fil directeur de nos travaux. Nous nous intéressons d'abord à une généralisation du seuil des répétitions (nous donnons des encadrements de sa valeur). Cette notion permet d'ajouter, pour décrire l'ensemble des répétitions à éviter, au paramètre de l'exposant, celui de la longueur des répétitions. Puis, nous étudions des problèmes d'existence de mots dans lesquels, simultanément, certaines répétitions sont interdites et d'autres sont forcées. Nous répondons, pour l'alphabet ternaire, à la question : quels réels sont l'exposant critique d'un mot infini sur un alphabet fixé? Nous introduisons ensuite une notion de haute répétitivité, et établissons une description partielle des couples d'exposants paramètrant une double contrainte de haute répétitivité et d'évitabilité. Pour finir, nous utilisons des résultats et techniques issus de ces problématiques pour résoudre une question de coloration de graphes : nous introduisons un seuil des répétitions, calqué sur celui connu pour les mots, et donnons sa valeur pour deux classes de graphes, les arbres et les graphes de subdivisions.In this thesis we study various problems on repetition avoidance in infinite words. Raised by Thue and motivated by his work on squarefree words, the topic developed during the 20th century, and has nowadays become a principal area of research in combinatorics on words. In 1972, Dejean proposed an important conjecture whose verification in steps was completed recently (2009). The conjecture concerns the repetition threshold for an alphabet, i.e., the infimum of the avoidable exponents for that alphabet. The notion of threshold as a borderline between avoidability and unavoidability for a given set of words is the guiding line of our work. First, we focus on a generalization of the repetition threshold. This concept allows us to include, in addition to the exponent, the length of the repetitions as a parameter in the description of the set of repetitions to avoid. We obtain various bounds in that respect. We then study existence problems for words in which simultaneously some repetitions are forbidden, and others are forced. For the ternary alphabet, we answer the question: what real numbers are the critical exponent of some infinite word over a given alphabet? Also, we introduce a notion of highly repetitive words and give a partial description of the pairs of exponents which parameterize the existence of words both highly repetitive and repetition-free. Finally, we use results and techniques stemming from those problems to solve a question on graph colouring: we introduce a repetition threshold adapted from the thresholds we know for words, and give its value for two classes of graphs, namely, trees and subdivision graphs.AIX-MARSEILLE2-Bib.electronique (130559901) / SudocSudocFranceF

    Around the p-adic Littlewood Conjecture

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    In this dissertation, we look at the Littlewood Conjecture and several related open problems. We introduce notions and theorems from various fields in order to properly formulate the conjectures and properly state the various results related to them. In Chapter 5, we investigate a potential counter-example to the p-adic Littlewood conjecture when p=2 via an intricate construction, and show that this potential counter-example does indeed satisfy the premises of the conjecture

    Subject index volumes 1–92

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