18 research outputs found
Nonrepetitive colorings of lexicographic product of graphs
A coloring of the vertices of a graph is nonrepetitive if there
exists no path for which for all
. Given graphs and with , the lexicographic
product is the graph obtained by substituting every vertex of by a
copy of , and every edge of by a copy of . %Our main results
are the following. We prove that for a sufficiently long path , a
nonrepetitive coloring of needs at least
colors. If then we need exactly colors to nonrepetitively color
, where is the empty graph on vertices. If we further require
that every copy of be rainbow-colored and the path is sufficiently
long, then the smallest number of colors needed for is at least
and at most . Finally, we define fractional nonrepetitive
colorings of graphs and consider the connections between this notion and the
above results
Nonrepetitive colorings of lexicographic product of paths and other graphs
A coloring of the vertices of a graph is nonrepetitive if
there exists no path for which
for all . Given graphs and
with , the lexicographic product is the graph
obtained by substituting every vertex of by a copy of , and
every edge of by a copy of .
We prove that for a sufficiently long path , a nonrepetitive
coloring of needs at least
colors. If then we need exactly colors to
nonrepetitively color , where is the empty graph on
vertices. If we further require that every copy of be
rainbow-colored and the path is sufficiently long, then the
smallest number of colors needed for is at least and
at most . Finally, we define fractional
nonrepetitive colorings of graphs and consider the connections
between this notion and the above results
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
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