27 research outputs found
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
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
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
A note on the Thue chromatic number of lexicographic products of graphs
A sequence is called non-repetitive if none of its subsequences forms a
repetition (a sequence r1r2 · · · r2n such that ri = rn+i for all 1 ≤ i ≤ n). Let G be a graph whose vertices are coloured. A colouring ϕ of the graph G is non-repetitive if the sequence of colours on every path in G is non-repetitive. The Thue chromatic number, denoted by π(G), is the minimum number of colours of a non-repetitive colouring of G