Results 1 
6 of
6
Splitting necklaces and measurable colorings of the real line
, 2009
"... A (continuous) necklace is simply an interval of the real line colored measurably with some number of colors. A wellknown application of the BorsukUlam theorem asserts that every kcolored necklace can be fairly split by at most k cuts (from the resulting pieces one can form two collections, eac ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
A (continuous) necklace is simply an interval of the real line colored measurably with some number of colors. A wellknown application of the BorsukUlam theorem asserts that every kcolored necklace can be fairly split by at most k cuts (from the resulting pieces one can form two collections, each capturing the same measure of every color). Here we prove that for every k ≥ 1 there is a measurable (k+3)coloring of the real line such that no interval can be fairly split using at most k cuts. In particular, there is a measurable 4coloring of the real line in which no two adjacent intervals have the same measure of every color. An analogous problem for the integers was posed by Erdős in 1961 and solved in the affirmative by Keränen in 1991. Curiously, in the discrete case the desired coloring also uses four colors.
On ternary squarefree circular words
"... Circular words are cyclically ordered finite sequences of letters. We give a computerfree proof of the following result by Currie: squarefree circular words over the ternary alphabet exist for all lengths l except for 5, 7, 9, 10, 14, and 17. Our proof reveals an interesting connection between ter ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
Circular words are cyclically ordered finite sequences of letters. We give a computerfree proof of the following result by Currie: squarefree circular words over the ternary alphabet exist for all lengths l except for 5, 7, 9, 10, 14, and 17. Our proof reveals an interesting connection between ternary squarefree circular words and closed walks in the K3,3 graph. In addition, our proof implies an exponential lower bound on the number of such circular words of length l and allows one to list all lengths l for which such a circular word is unique up to isomorphism.
September 2428, 2012 Nonrepetitive, acyclic and clique colorings of graphs with few P 4 's
"... Abstract In this paper, we propose algorithms to determine the Thue chromatic number and the clique chromatic number of P 4 tidy graphs and (q, q − 4)graphs. These classes include cographs and P 4 sparse graphs. All algorithms have lineartime complexity, for fixed q, and then are fixed paramete ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract In this paper, we propose algorithms to determine the Thue chromatic number and the clique chromatic number of P 4 tidy graphs and (q, q − 4)graphs. These classes include cographs and P 4 sparse graphs. All algorithms have lineartime complexity, for fixed q, and then are fixed parameter tractable. All these coloring problems are known to be NPhard for general graphs. We also prove that every connected (q, q − 4)graph with at least q vertices is 2cliquecolorable and that every acyclic coloring of a cograph is also nonrepetitive, generalizing a result from
REPETITION THRESHOLDS FOR SUBDIVIDED GRAPHS AND TREES
 THEORETICAL INFORMATICS AND APPLICATIONS
, 1999
"... The repetition threshold introduced by Dejean and Brandenburg is the smallest real number α such that there exists an infinite word over a kletter alphabet that avoids βpowers for all β> α. We extend this notion to colored graphs and obtain the value of the repetition thresholds of trees and “ ..."
Abstract
 Add to MetaCart
The repetition threshold introduced by Dejean and Brandenburg is the smallest real number α such that there exists an infinite word over a kletter alphabet that avoids βpowers for all β> α. We extend this notion to colored graphs and obtain the value of the repetition thresholds of trees and “large enough” subdivisions of graphs for every alphabet size.