29 research outputs found
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
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
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