3,488 research outputs found
Recoloring graphs via tree decompositions
Let be an integer. Two vertex -colorings of a graph are
\emph{adjacent} if they differ on exactly one vertex. A graph is
\emph{-mixing} if any proper -coloring can be transformed into any other
through a sequence of adjacent proper -colorings. Jerrum proved that any
graph is -mixing if is at least the maximum degree plus two. We first
improve Jerrum's bound using the grundy number, which is the worst number of
colors in a greedy coloring.
Any graph is -mixing, where is the treewidth of the graph
(Cereceda 2006). We prove that the shortest sequence between any two
-colorings is at most quadratic (which is optimal up to a constant
factor), a problem left open in Bonamy et al. (2012).
We also prove that given any two -colorings of a cograph (resp.
distance-hereditary graph) , we can find a linear (resp. quadratic) sequence
between them. In both cases, the bounds cannot be improved by more than a
constant factor for a fixed . The graph classes are also optimal in
some sense: one of the smallest interesting superclass of distance-hereditary
graphs corresponds to comparability graphs, for which no such property holds
(even when relaxing the constraint on the length of the sequence). As for
cographs, they are equivalently the graphs with no induced , and there
exist -free graphs that admit no sequence between two of their
-colorings.
All the proofs are constructivist and lead to polynomial-time recoloring
algorithmComment: 20 pages, 8 figures, partial results already presented in
http://arxiv.org/abs/1302.348
Three-coloring triangle-free graphs on surfaces V. Coloring planar graphs with distant anomalies
We settle a problem of Havel by showing that there exists an absolute
constant d such that if G is a planar graph in which every two distinct
triangles are at distance at least d, then G is 3-colorable. In fact, we prove
a more general theorem. Let G be a planar graph, and let H be a set of
connected subgraphs of G, each of bounded size, such that every two distinct
members of H are at least a specified distance apart and all triangles of G are
contained in \bigcup{H}. We give a sufficient condition for the existence of a
3-coloring phi of G such that for every B\in H, the restriction of phi to B is
constrained in a specified way.Comment: 26 pages, no figures. Updated presentatio
Recoloring bounded treewidth graphs
Let be an integer. Two vertex -colorings of a graph are
\emph{adjacent} if they differ on exactly one vertex. A graph is
\emph{-mixing} if any proper -coloring can be transformed into any other
through a sequence of adjacent proper -colorings. Any graph is
-mixing, where is the treewidth of the graph (Cereceda 2006). We
prove that the shortest sequence between any two -colorings is at most
quadratic, a problem left open in Bonamy et al. (2012).
Jerrum proved that any graph is -mixing if is at least the maximum
degree plus two. We improve Jerrum's bound using the grundy number, which is
the worst number of colors in a greedy coloring.Comment: 11 pages, 5 figure
Subdivision into i-packings and S-packing chromatic number of some lattices
An -packing in a graph is a set of vertices at pairwise distance
greater than . For a nondecreasing sequence of integers
, the -packing chromatic number of a graph is
the least integer such that there exists a coloring of into colors
where each set of vertices colored , , is an -packing.
This paper describes various subdivisions of an -packing into -packings
(j\textgreater{}i) for the hexagonal, square and triangular lattices. These
results allow us to bound the -packing chromatic number for these graphs,
with more precise bounds and exact values for sequences ,
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