17 research outputs found
Planar graphs have bounded nonrepetitive chromatic number
A colouring of a graph isnonrepetitiveif for every path of even order, thesequence of colours on the first half of the path is different from the sequence of colours onthe second half. We show that planar graphs have nonrepetitive colourings with a boundednumber of colours, thus proving a conjecture of Alon, Grytczuk, Hałuszczak and Riordan(2002). We also generalise this result for graphs of bounded Euler genus, graphs excluding afixed minor, and graphs excluding a fixed topological minor
Nonrepetitive Colourings of Planar Graphs with Colours
A vertex colouring of a graph is \emph{nonrepetitive} if there is no path for
which the first half of the path is assigned the same sequence of colours as
the second half. The \emph{nonrepetitive chromatic number} of a graph is
the minimum integer such that has a nonrepetitive -colouring.
Whether planar graphs have bounded nonrepetitive chromatic number is one of the
most important open problems in the field. Despite this, the best known upper
bound is for -vertex planar graphs. We prove a
upper bound
Nonrepetitive colourings of planar graphs with O(log n) colours
A vertex colouring of a graph is nonrepetitive if there is no path for which the first half of the path is assigned the same sequence of colours as the second half. The nonrepetitive chromatic number of a graph G is the minimum integer k such that G has a nonrepetitive k-colouring. Whether planar graphs have bounded nonrepetitive chromatic number is one of the most important open problems in the field. Despite this, the best known upper bound is O(n−−√) for n-vertex planar graphs. We prove a O(logn) upper bound
A note about online nonrepetitive coloring -trees
We prove that it is always possible to color online nonrepetitively any
(partial) -tree (that is, graphs with tree-width at most ) with
colors. This implies that it is always possible to color online nonrepetitively
cycles, trees and series-parallel graphs with colors. Our results
generalize the respective (offline) nonrepetitive coloring results
The Weak Circular Repetition Threshold Over Large Alphabets
The repetition threshold for words on letters, denoted \mbox{RT}(n), is
the infimum of the set of all such that there are arbitrarily long -free
words over letters. A repetition threshold for circular words on
letters can be defined in three natural ways, which gives rise to the weak,
intermediate, and strong circular repetition thresholds for letters,
denoted \mbox{CRT}_{\mbox{W}}(n), \mbox{CRT}_{\mbox{I}}(n), and
\mbox{CRT}_{\mbox{S}}(n), respectively. Currie and the present authors
conjectured that
\mbox{CRT}_{\mbox{I}}(n)=\mbox{CRT}_{\mbox{W}}(n)=\mbox{RT}(n) for all . We prove that \mbox{CRT}_{\mbox{W}}(n)=\mbox{RT}(n) for all ,
which confirms a weak version of this conjecture for all but finitely many
values of .Comment: arXiv admin note: text overlap with arXiv:1911.0577
Another approach to non-repetitive colorings of graphs of bounded degree
We propose a new proof technique that aims to be applied to the same problems
as the Lov\'asz Local Lemma or the entropy-compression method. We present this
approach in the context of non-repetitive colorings and we use it to improve
upper-bounds relating different non-repetitive numbers to the maximal degree of
a graph. It seems that there should be other interesting applications to the
presented approach.
In terms of upper-bound our approach seems to be as strong as
entropy-compression, but the proofs are more elementary and shorter. The
application we provide in this paper are upper bounds for graphs of maximal
degree at most : a minor improvement on the upper-bound of the
non-repetitive number, a upper-bound on the weak total
non-repetitive number and a
upper-bound on the total non-repetitive number of graphs. This last result
implies the same upper-bound for the non-repetitive index of graphs, which
improves the best known bound
Clustered 3-Colouring Graphs of Bounded Degree
A (not necessarily proper) vertex colouring of a graph has "clustering"
if every monochromatic component has at most vertices. We prove that planar
graphs with maximum degree are 3-colourable with clustering
. The previous best bound was . This result for
planar graphs generalises to graphs that can be drawn on a surface of bounded
Euler genus with a bounded number of crossings per edge. We then prove that
graphs with maximum degree that exclude a fixed minor are 3-colourable
with clustering . The best previous bound for this result was
exponential in .Comment: arXiv admin note: text overlap with arXiv:1904.0479