Nonrepetitive colorings of line arrangements
Jarosław Grytczuk , Karol Kosiński , Michał Zmarz
Abstract
A sequence is nonrepetitive if no two adjacent segments of are identical. A famous result of Thue from 1906 asserts that there are arbitrarily long nonrepetitive sequences over 3 symbols. We study the following geometric variant of this problem. Given a set of points in the plane and a set of lines, what is the least number of colors needed to color so that every line in is nonrepetitive? If consists of all intersection points of a prescribed set of lines , then we prove that there is such coloring using at most 405 colors. The proof is based on a theorem of Thue and on a result of Alon and Marshall concerning homomorphisms of edge colored planar graphs. We also consider nonrepetitive colorings involving other geometric structures. For instance, a nonrepetitive analog of the famous Hadwiger–Nelson problem is formulated as follows: what is the least number of colors needed to color the plane so that every path of the unit distance graph whose vertices are colinear is nonrepetitive? Using a theorem of Thue we prove that this number is at most .Author | |
Journal series | European Journal of Combinatorics, ISSN 0195-6698 |
Issue year | 2016 |
Vol | 51 |
Pages | 275-279 |
Publication size in sheets | 0.5 |
ASJC Classification | |
DOI | DOI:10.1016/j.ejc.2015.05.013 |
URL | https://www.sciencedirect.com/science/article/pii/S0195669815001304?via%3Dihub |
Language | en angielski |
Score (nominal) | 30 |
Score source | journalList |
Score | = 25.0, 09-01-2020, ArticleFromJournal = 30.0, 09-01-2020, ArticleFromJournal |
Publication indicators | = 4; = 3; : 2016 = 1.167; : 2016 = 0.786 (2) - 2016=0.828 (5) |
Citation count* |
* presented citation count is obtained through Internet information analysis and it is close to the number calculated by the Publish or Perish system.
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