Contributions to combinatorics on words in an abelian context and covering problems in graphs

Abstract : This dissertation is divided into two (distinct but connected) parts that reflect the joint PhD. We study and we solve several questions regarding on the one hand combinatorics on words in an abelian context and on the other hand covering problems in graphs. Each particular problem is the topic of a chapter. In combinatorics on words, the first problem considered focuses on the 2-regularity of sequences in the sense of Allouche and Shallit. We prove that a sequence satisfying a certain symmetry property is 2-regular. Then we apply this theorem to show that the 2-abelian complexity functions of the Thue--Morse word and the period-doubling word are 2-regular. The computation and arguments leading to these results fit into a quite general scheme that we hope can be used again to prove additional regularity results. The second question concerns the notion of return words up to abelian equivalence, introduced by Puzynina and Zamboni. We obtain a characterization of Sturmian words with non-zero intercept in terms of the finiteness of the set of abelian return words to all prefixes. We describe this set of abelian returns for the Fibonacci word but also for the Thue-Morse word (which is not Sturmian). We investigate the relationship existing between the abelian complexity and the finiteness of this set. In graph theory, the first problem considered deals with identifying codes in graphs. These codes were introduced by Karpovsky, Chakrabarty and Levitin to model fault-diagnosis in multiprocessor systems. The ratio between the optimal size of an identifying code and the optimal size of a fractional relaxation of an identifying code is between 1 and 2 ln(|V|)+1 where V is the vertex set of the graph. We focus on vertex-transitive graphs, since we can compute the exact fractional solution for them. We exhibit infinite families, called generalized quadrangles, of vertex-transitive graphs with integer and fractional identifying codes of order |V|^k with k in {1/4,1/3,2/5}. The second problem concerns (r,a,b)-covering codes of the infinite grid already studied by Axenovich and Puzynina. We introduce the notion of constant 2-labellings of weighted graphs and study them in four particular weighted cycles. We present a method to link these labellings with covering codes. Finally, we determine the precise values of the constants a and b of any (r,a,b)-covering code of the infinite grid with |a-b|>4. This is an extension of a theorem of Axenovich.
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Elise Vandomme. Contributions to combinatorics on words in an abelian context and covering problems in graphs. General Mathematics [math.GM]. Université Grenoble Alpes, 2015. English. ⟨NNT : 2015GREAM010⟩. ⟨tel-01162090⟩

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