1,879 research outputs found
The Fractional Chromatic Number of the Plane
The chromatic number of the plane is the chromatic number of the uncountably
infinite graph that has as its vertices the points of the plane and has an edge
between two points if their distance is 1. This chromatic number is denoted
. The problem was introduced in 1950, and shortly
thereafter it was proved that . These bounds are
both easy to prove, but after more than 60 years they are still the best known.
In this paper, we investigate , the fractional chromatic
number of the plane. The previous best bounds (rounded to five decimal places)
were . Here we improve the lower
bound to .Comment: 20 pages, 10 figure
Planar graphs are 9/2-colorable
We show that every planar graph has a 2-fold 9-coloring. In particular,
this implies that has fractional chromatic number at most . This
is the first proof (independent of the 4 Color Theorem) that there exists a
constant such that every planar has fractional chromatic number at
most .Comment: 12 pages, 6 figures; following the suggestion of an editor, we split
the original version of this paper into two papers: one is the current
version of this paper, and the other is "Planar graphs have Independence
Ratio at least 3/13" (also available on the arXiv
Making Octants Colorful and Related Covering Decomposition Problems
We give new positive results on the long-standing open problem of geometric
covering decomposition for homothetic polygons. In particular, we prove that
for any positive integer k, every finite set of points in R^3 can be colored
with k colors so that every translate of the negative octant containing at
least k^6 points contains at least one of each color. The best previously known
bound was doubly exponential in k. This yields, among other corollaries, the
first polynomial bound for the decomposability of multiple coverings by
homothetic triangles. We also investigate related decomposition problems
involving intervals appearing on a line. We prove that no algorithm can
dynamically maintain a decomposition of a multiple covering by intervals under
insertion of new intervals, even in a semi-online model, in which some coloring
decisions can be delayed. This implies that a wide range of sweeping plane
algorithms cannot guarantee any bound even for special cases of the octant
problem.Comment: version after revision process; minor changes in the expositio
On the multiple Borsuk numbers of sets
The Borsuk number of a set S of diameter d >0 in Euclidean n-space is the
smallest value of m such that S can be partitioned into m sets of diameters
less than d. Our aim is to generalize this notion in the following way: The
k-fold Borsuk number of such a set S is the smallest value of m such that there
is a k-fold cover of S with m sets of diameters less than d. In this paper we
characterize the k-fold Borsuk numbers of sets in the Euclidean plane, give
bounds for those of centrally symmetric sets, smooth bodies and convex bodies
of constant width, and examine them for finite point sets in the Euclidean
3-space.Comment: 16 pages, 3 figure
Learning from networked examples
Many machine learning algorithms are based on the assumption that training
examples are drawn independently. However, this assumption does not hold
anymore when learning from a networked sample because two or more training
examples may share some common objects, and hence share the features of these
shared objects. We show that the classic approach of ignoring this problem
potentially can have a harmful effect on the accuracy of statistics, and then
consider alternatives. One of these is to only use independent examples,
discarding other information. However, this is clearly suboptimal. We analyze
sample error bounds in this networked setting, providing significantly improved
results. An important component of our approach is formed by efficient sample
weighting schemes, which leads to novel concentration inequalities
Fractional Analogues in Graph Theory
Tait showed in 1878 that the Four Color Theorem is equivalent to being able to three-color the edges of any planar, three-regular, two-edge connected graph. Not surprisingly, this equivalent problem proved to be equally difficult. We consider the problem of fractional colorings, which resemble ordinary colorings but allow for some degree of cheating. Happily, it is known that every planar three-regular, two-edge connected graph is fractionally three-edge colorable. Is there an analogue to Tait’s Theorem which would allow us to derive the Fractional Four Color Theorem from this edge-coloring result
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