ПЕРИОДИЧЕСКИЕ СТРУКТУРЫ В МОРФИЧЕСКИХ СЛОВАХ И РАСКРАСКАХ БЕСКОНЕЧНЫХ ЦИРКУЛЯНТНЫХ ГРАФОВ

Abstract : The content of the thesis is comprised of two parts: one deals with combinatorial properties of infinite words and the other with graph coloring problems.The first main part of the manuscript concerns regular structures in infinite aperiodic words, such as arithmetic subsequences and complete first returns.We study the function that outputs the maximal length of a monochromatic arithmetic subsequence (an arithmetic progression) as a function of the common difference d for a family of uniform morphic words, which includes the Thue-Morse word. We obtain the explicit upper bound on the rate of growth of the function and locations of arithmetic progressions of maximal lengths and difference d. To study periodic arithmetic subsequences in infinite words we define the notion of an arithmetic index and obtain upper and lower bounds on the rate of growth of the function of arithmetic index in the same family of words.Another topic in this direction involves the study of two new complexity functions of infinite words based on the notions of open and closed words. We derive explicit formulae for the open and closed complexity functions for an Arnoux-Rauzy word over an alphabet of finite cardinality.The second main part of the thesis deals with perfect colorings (a.k.a. equitable partitions) of infinite graphs of bounded degree. We study Caley graphs of infinite additive groups with a prescribed set of generators. We consider the case when the set of generators is composed of integers from the interval [-n,n], and the case when the generators are odd integers from [-2n-1,2n+1], where n is a positive integer. For both families of graphs, we obtain a complete characterization of perfect 2-colorings
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Olga Parshina. ПЕРИОДИЧЕСКИЕ СТРУКТУРЫ В МОРФИЧЕСКИХ СЛОВАХ И РАСКРАСКАХ БЕСКОНЕЧНЫХ ЦИРКУЛЯНТНЫХ ГРАФОВ. Combinatorics [math.CO]. Université de Lyon; Sobolev Institute of Mathematics, 2019. Russian. ⟨NNT : 2019LYSE1071⟩. ⟨tel-02301714⟩

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