A new approach to nonrepetitive sequences

A sequence is nonrepetitive if it does not contain two adjacent identical blocks. The remarkable construction of Thue asserts that 3 symbols are enough to build an arbitrarily long nonrepetitive sequence. It is still not settled whether the following extension holds: for every sequence of 3-element sets $L_1,..., L_n$ there exists a nonrepetitive sequence $s_1, ..., s_n$ with $s_i\in L_i$. Applying the probabilistic method one can prove that this is true for sufficiently large sets $L_i$. We present an elementary proof that sets of size 4 suffice (confirming the best known bound). The argument is a simple counting with Catalan numbers involved. Our approach is inspired by a new algorithmic proof of the Lov\'{a}sz Local Lemma due to Moser and Tardos and its interpretations by Fortnow and Tao. The presented method has further applications to nonrepetitive games and nonrepetitive colorings of graphs.Comment: 5 pages, no figures.arXiv admin note: substantial text overlap with arXiv:1103.381

On the facial Thue choice index via entropy compression

A sequence is nonrepetitive if it contains no identical consecutive subsequences. An edge colouring of a path is nonrepetitive if the sequence of colours of its consecutive edges is nonrepetitive. By the celebrated construction of Thue, it is possible to generate nonrepetitive edge colourings for arbitrarily long paths using only three colours. A recent generalization of this concept implies that we may obtain such colourings even if we are forced to choose edge colours from any sequence of lists of size 4 (while sufficiency of lists of size 3 remains an open problem). As an extension of these basic ideas, Havet, Jendrol', Sot\'ak and \v{S}krabul'\'akov\'a proved that for each plane graph, 8 colours are sufficient to provide an edge colouring so that every facial path is nonrepetitively coloured. In this paper we prove that the same is possible from lists, provided that these have size at least 12. We thus improve the previous bound of 291 (proved by means of the Lov\'asz Local Lemma). Our approach is based on the Moser-Tardos entropy-compression method and its recent extensions by Grytczuk, Kozik and Micek, and by Dujmovi\'c, Joret, Kozik and Wood

Online version of the theorem of Thue

A sequence S is nonrepetitive if no two adjacent blocks of S are the same. In 1906 Thue proved that there exist arbitrarily long nonrepetitive sequences over 3 symbols. We consider the online variant of this result in which a nonrepetitive sequence is constructed during a play between two players: Bob is choosing a position in a sequence and Alice is inserting a symbol on that position taken from a fixed set A. The goal of Bob is to force Alice to create a repetition, and if he succeeds, then the game stops. The goal of Alice is naturally to avoid that and thereby to construct a nonrepetitive sequence of any given length. We prove that Alice has a strategy to play arbitrarily long provided the size of the set A is at least 12. This is the online version of the Theorem of Thue. The proof is based on nonrepetitive colorings of outerplanar graphs. On the other hand, one can prove that even over 4 symbols Alice has no chance to play for too long. The minimum size of the set of symbols needed for the online version of Thue's theorem remains unknown

New Bounds for Facial Nonrepetitive Colouring

We prove that the facial nonrepetitive chromatic number of any outerplanar graph is at most 11 and of any planar graph is at most 22.Comment: 16 pages, 5 figure

Pathwidth and nonrepetitive list coloring

A vertex coloring of a graph is nonrepetitive if there is no path in the graph whose first half receives the same sequence of colors as the second half. While every tree can be nonrepetitively colored with a bounded number of colors (4 colors is enough), Fiorenzi, Ochem, Ossona de Mendez, and Zhu recently showed that this does not extend to the list version of the problem, that is, for every $\ell \geq 1$ there is a tree that is not nonrepetitively $\ell$-choosable. In this paper we prove the following positive result, which complements the result of Fiorenzi et al.: There exists a function $f$ such that every tree of pathwidth $k$ is nonrepetitively $f(k)$-choosable. We also show that such a property is specific to trees by constructing a family of pathwidth-2 graphs that are not nonrepetitively $\ell$-choosable for any fixed $\ell$.Comment: v2: Minor changes made following helpful comments by the referee

Genomic Selective Constraints in Murid Noncoding DNA

Recent work has suggested that there are many more selectively constrained, functional noncoding than coding sites in mammalian genomes. However, little is known about how selective constraint varies amongst different classes of noncoding DNA. We estimated the magnitude of selective constraint on a large dataset of mouse-rat gene orthologs and their surrounding noncoding DNA. Our analysis indicates that there are more than three times as many selectively constrained, nonrepetitive sites within noncoding DNA as in coding DNA in murids. The majority of these constrained noncoding sites appear to be located within intergenic regions, at distances greater than 5 kilobases from known genes. Our study also shows that in murids, intron length and mean intronic selective constraint are negatively correlated with intron ordinal number. Our results therefore suggest that functional intronic sites tend to accumulate toward the 5' end of murid genes. Our analysis also reveals that mean number of selectively constrained noncoding sites varies substantially with the function of the adjacent gene. We find that, among others, developmental and neuronal genes are associated with the greatest numbers of putatively functional noncoding sites compared with genes involved in electron transport and a variety of metabolic processes. Combining our estimates of the total number of constrained coding and noncoding bases we calculate that over twice as many deleterious mutations have occurred in intergenic regions as in known genic sequence and that the total genomic deleterious point mutation rate is 0.91 per diploid genome, per generation. This estimated rate is over twice as large as a previous estimate in murids