3,090 research outputs found
Data Reduction for Graph Coloring Problems
This paper studies the kernelization complexity of graph coloring problems
with respect to certain structural parameterizations of the input instances. We
are interested in how well polynomial-time data reduction can provably shrink
instances of coloring problems, in terms of the chosen parameter. It is well
known that deciding 3-colorability is already NP-complete, hence parameterizing
by the requested number of colors is not fruitful. Instead, we pick up on a
research thread initiated by Cai (DAM, 2003) who studied coloring problems
parameterized by the modification distance of the input graph to a graph class
on which coloring is polynomial-time solvable; for example parameterizing by
the number k of vertex-deletions needed to make the graph chordal. We obtain
various upper and lower bounds for kernels of such parameterizations of
q-Coloring, complementing Cai's study of the time complexity with respect to
these parameters.
Our results show that the existence of polynomial kernels for q-Coloring
parameterized by the vertex-deletion distance to a graph class F is strongly
related to the existence of a function f(q) which bounds the number of vertices
which are needed to preserve the NO-answer to an instance of q-List-Coloring on
F.Comment: Author-accepted manuscript of the article that will appear in the FCT
2011 special issue of Information & Computatio
5-list-coloring planar graphs with distant precolored vertices
We answer positively the question of Albertson asking whether every planar
graph can be -list-colored even if it contains precolored vertices, as long
as they are sufficiently far apart from each other. In order to prove this
claim, we also give bounds on the sizes of graphs critical with respect to
5-list coloring. In particular, if G is a planar graph, H is a connected
subgraph of G and L is an assignment of lists of colors to the vertices of G
such that |L(v)| >= 5 for every v in V(G)-V(H) and G is not L-colorable, then G
contains a subgraph with O(|H|^2) vertices that is not L-colorable.Comment: 53 pages, 9 figures version 2: addresses suggestions by reviewer
Three-coloring triangle-free graphs on surfaces III. Graphs of girth five
We show that the size of a 4-critical graph of girth at least five is bounded
by a linear function of its genus. This strengthens the previous bound on the
size of such graphs given by Thomassen. It also serves as the basic case for
the description of the structure of 4-critical triangle-free graphs embedded in
a fixed surface, presented in a future paper of this series.Comment: 53 pages, 7 figures; updated according to referee remark
Testing first-order properties for subclasses of sparse graphs
We present a linear-time algorithm for deciding first-order (FO) properties
in classes of graphs with bounded expansion, a notion recently introduced by
Nesetril and Ossona de Mendez. This generalizes several results from the
literature, because many natural classes of graphs have bounded expansion:
graphs of bounded tree-width, all proper minor-closed classes of graphs, graphs
of bounded degree, graphs with no subgraph isomorphic to a subdivision of a
fixed graph, and graphs that can be drawn in a fixed surface in such a way that
each edge crosses at most a constant number of other edges. We deduce that
there is an almost linear-time algorithm for deciding FO properties in classes
of graphs with locally bounded expansion.
More generally, we design a dynamic data structure for graphs belonging to a
fixed class of graphs of bounded expansion. After a linear-time initialization
the data structure allows us to test an FO property in constant time, and the
data structure can be updated in constant time after addition/deletion of an
edge, provided the list of possible edges to be added is known in advance and
their simultaneous addition results in a graph in the class. All our results
also hold for relational structures and are based on the seminal result of
Nesetril and Ossona de Mendez on the existence of low tree-depth colorings
Pathwidth and nonrepetitive list coloring
A vertex coloring of a graph is nonrepetitive if there is no path in the
graph whose first half receives the same sequence of colors as the second half.
While every tree can be nonrepetitively colored with a bounded number of colors
(4 colors is enough), Fiorenzi, Ochem, Ossona de Mendez, and Zhu recently
showed that this does not extend to the list version of the problem, that is,
for every there is a tree that is not nonrepetitively
-choosable. In this paper we prove the following positive result, which
complements the result of Fiorenzi et al.: There exists a function such
that every tree of pathwidth is nonrepetitively -choosable. We also
show that such a property is specific to trees by constructing a family of
pathwidth-2 graphs that are not nonrepetitively -choosable for any fixed
.Comment: v2: Minor changes made following helpful comments by the referee
Independent transversals in locally sparse graphs
Let G be a graph with maximum degree \Delta whose vertex set is partitioned
into parts V(G) = V_1 \cup ... \cup V_r. A transversal is a subset of V(G)
containing exactly one vertex from each part V_i. If it is also an independent
set, then we call it an independent transversal. The local degree of G is the
maximum number of neighbors of a vertex v in a part V_i, taken over all choices
of V_i and v \not \in V_i. We prove that for every fixed \epsilon > 0, if all
part sizes |V_i| >= (1+\epsilon)\Delta and the local degree of G is o(\Delta),
then G has an independent transversal for sufficiently large \Delta. This
extends several previous results and settles (in a stronger form) a conjecture
of Aharoni and Holzman. We then generalize this result to transversals that
induce no cliques of size s. (Note that independent transversals correspond to
s=2.) In that context, we prove that parts of size |V_i| >=
(1+\epsilon)[\Delta/(s-1)] and local degree o(\Delta) guarantee the existence
of such a transversal, and we provide a construction that shows this is
asymptotically tight.Comment: 16 page
Beyond Ohba's Conjecture: A bound on the choice number of -chromatic graphs with vertices
Let denote the choice number of a graph (also called "list
chromatic number" or "choosability" of ). Noel, Reed, and Wu proved the
conjecture of Ohba that when . We
extend this to a general upper bound: . Our result is sharp for
using Ohba's examples, and it improves the best-known
upper bound for .Comment: 14 page
Testing first-order properties for subclasses of sparse graphs
We present a linear-time algorithm for deciding first-order (FO) properties in classes of graphs with bounded expansion, a notion recently introduced by Nešetřil and Ossona de Mendez. This generalizes several results from the literature, because many natural classes of graphs have bounded expansion: graphs of bounded tree-width, all proper minor-closed classes of graphs, graphs of bounded degree, graphs with no subgraph isomorphic to a subdivision of a fixed graph, and graphs that can be drawn in a fixed surface in such a way that each edge crosses at most a constant number of other edges. We deduce that there is an almost linear-time algorithm for deciding FO properties in classes of graphs with locally bounded expansion.
More generally, we design a dynamic data structure for graphs belonging to a fixed class of graphs of bounded expansion. After a linear-time initialization the data structure allows us to test an FO property in constant time, and the data structure can be updated in constant time after addition/deletion of an edge, provided the list of possible edges to be added is known in advance and their simultaneous addition results in a graph in the class. All our results also hold for relational structures and are based on the seminal result of Nešetřil and Ossona de Mendez on the existence of low tree-depth colorings
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