2,907 research outputs found
Max Dehn, Axel Thue, and the Undecidable
This is a short essay on the roles of Max Dehn and Axel Thue in the
formulation of the word problem for (semi)groups, and the story of the proofs
showing that the word problem is undecidable.Comment: Definition of undecidability and unsolvability improve
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 there is a tree that is not nonrepetitively
-choosable. In this paper we prove the following positive result, which
complements the result of Fiorenzi et al.: There exists a function such
that every tree of pathwidth is nonrepetitively -choosable. We also
show that such a property is specific to trees by constructing a family of
pathwidth-2 graphs that are not nonrepetitively -choosable for any fixed
.Comment: v2: Minor changes made following helpful comments by the referee
Thue's 1914 paper: a translation
This paper includes notes to accompany a reading of Thue's 1914 paper
"Probleme uber Veranderungen von Zeichenreihen nach gegebenen Reglen", along
with a translation of that paper. Thue's 1914 paper is mainly famous for
proving an early example of an undecidable problem, cited prominently by Post.
However, Post's paper principally makes use of the definition of Thue systems,
described on the first two pages of Thue's paper, and does not depend on the
more specific results in the remainder of Thue's paper. A closer study of the
remaining parts of that paper highlight a number of important themes in the
history of computing: the transition from algebra to formal language theory,
the analysis of the "computational power" (in a pre-1936 sense) of rules, and
the development of algorithms to generate rule-sets
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
Aperiodic colorings and tilings of Coxeter groups
We construct a limit aperiodic coloring of hyperbolic groups. Also we
construct limit strongly aperiodic strictly balanced tilings of the Davis
complex for all Coxeter groups
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