5,260 research outputs found
An NP-Complete Problem in Grid Coloring
A c-coloring of G(n,m)=n x m is a mapping of G(n,m) into {1,...,c} such that
no four corners forming a rectangle have the same color. In 2009 a challenge
was proposed via the internet to find a 4-coloring of G(17,17). This attracted
considerable attention from the popular mathematics community. A coloring was
produced; however, finding it proved to be difficult. The question arises: is
the problem of grid coloring is difficult in general? We present three results
that support this conjecture, (1) an NP completeness result, (2) a lower bound
on Tree-resolution, (3) a lower bound on Tree-CP proofs. Note that items (2)
and (3) yield statements from Ramsey Theory which are of size polynomial in
their parameters and require exponential size in various proof systems.Comment: 25 page
Node Coloring in Wireless Networks: Complexity Results and Grid Coloring
International audienceColoring is used in wireless networks to improve communication efficiency, mainly in terms of bandwidth, energy and possibly end-to-end delays. In this paper, we define the h-hop node coloring problem, with h any positive integer, adapted to two types of applications in wireless networks. We specify both general mode for general applications and strategic mode for data gathering applications.We prove that the associated decision problem is NP-complete. We then focus on grid topologies that constitute regular topologies for large or dense wireless networks. We consider various transmission ranges and identify a color pattern that can be reproduced to color the whole grid with the optimal number of colors. We obtain an optimal periodic coloring of the grid for the considered transmission range. We then present a 3-hop distributed coloring algorithm, called SERENA. Through simulation results, we highlight the impact of node priority assignment on the number of colors obtained for any network and grids in particular. We then compare these optimal results on grids with those obtained by SERENA and identify directions to improve SERENA
Grid Representations and the Chromatic Number
A grid drawing of a graph maps vertices to grid points and edges to line
segments that avoid grid points representing other vertices. We show that there
is a number of grid points that some line segment of an arbitrary grid drawing
must intersect. This number is closely connected to the chromatic number.
Second, we study how many columns we need to draw a graph in the grid,
introducing some new \NP-complete problems. Finally, we show that any planar
graph has a planar grid drawing where every line segment contains exactly two
grid points. This result proves conjectures asked by David Flores-Pe\~naloza
and Francisco Javier Zaragoza Martinez.Comment: 22 pages, 8 figure
Vertex-Coloring with Star-Defects
Defective coloring is a variant of traditional vertex-coloring, according to
which adjacent vertices are allowed to have the same color, as long as the
monochromatic components induced by the corresponding edges have a certain
structure. Due to its important applications, as for example in the
bipartisation of graphs, this type of coloring has been extensively studied,
mainly with respect to the size, degree, and acyclicity of the monochromatic
components.
In this paper we focus on defective colorings in which the monochromatic
components are acyclic and have small diameter, namely, they form stars. For
outerplanar graphs, we give a linear-time algorithm to decide if such a
defective coloring exists with two colors and, in the positive case, to
construct one. Also, we prove that an outerpath (i.e., an outerplanar graph
whose weak-dual is a path) always admits such a two-coloring. Finally, we
present NP-completeness results for non-planar and planar graphs of bounded
degree for the cases of two and three colors
Automatic frequency assignment for cellular telephones using constraint satisfaction techniques
We study the problem of automatic frequency assignment for cellular telephone
systems. The frequency assignment problem is viewed as the problem
to minimize the unsatisfied soft constraints in a constraint satisfaction problem
(CSP) over a finite domain of frequencies involving co-channel, adjacent
channel, and co-site constraints. The soft constraints are automatically derived
from signal strength prediction data. The CSP is solved using a generalized
graph coloring algorithm. Graph-theoretical results play a crucial
role in making the problem tractable. Performance results from a real-world
frequency assignment problem are presented.
We develop the generalized graph coloring algorithm by stepwise refinement,
starting from DSATUR and augmenting it with local propagation,
constraint lifting, intelligent backtracking, redundancy avoidance, and iterative
deepening
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