94,133 research outputs found
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
Balanced partitions of 3-colored geometric sets in the plane
Let SS be a finite set of geometric objects partitioned into classes or colors . A subset S'¿SS'¿S is said to be balanced if S'S' contains the same amount of elements of SS from each of the colors. We study several problems on partitioning 33-colored sets of points and lines in the plane into two balanced subsets: (a) We prove that for every 3-colored arrangement of lines there exists a segment that intersects exactly one line of each color, and that when there are 2m2m lines of each color, there is a segment intercepting mm lines of each color. (b) Given nn red points, nn blue points and nn green points on any closed Jordan curve ¿¿, we show that for every integer kk with 0=k=n0=k=n there is a pair of disjoint intervals on ¿¿ whose union contains exactly kk points of each color. (c) Given a set SS of nn red points, nn blue points and nn green points in the integer lattice satisfying certain constraints, there exist two rays with common apex, one vertical and one horizontal, whose union splits the plane into two regions, each one containing a balanced subset of SS.Peer ReviewedPostprint (published version
Comparing reverse complementary genomic words based on their distance distributions and frequencies
In this work we study reverse complementary genomic word pairs in the human
DNA, by comparing both the distance distribution and the frequency of a word to
those of its reverse complement. Several measures of dissimilarity between
distance distributions are considered, and it is found that the peak
dissimilarity works best in this setting. We report the existence of reverse
complementary word pairs with very dissimilar distance distributions, as well
as word pairs with very similar distance distributions even when both
distributions are irregular and contain strong peaks. The association between
distribution dissimilarity and frequency discrepancy is explored also, and it
is speculated that symmetric pairs combining low and high values of each
measure may uncover features of interest. Taken together, our results suggest
that some asymmetries in the human genome go far beyond Chargaff's rules. This
study uses both the complete human genome and its repeat-masked version.Comment: Post-print of a paper accepted to publication in "Interdisciplinary
Sciences: Computational Life Sciences" (ISSN: 1913-2751, ESSN: 1867-1462
Reverse mathematics and equivalents of the axiom of choice
We study the reverse mathematics of countable analogues of several maximality
principles that are equivalent to the axiom of choice in set theory. Among
these are the principle asserting that every family of sets has a
-maximal subfamily with the finite intersection property and the
principle asserting that if is a property of finite character then every
set has a -maximal subset of which holds. We show that these
principles and their variations have a wide range of strengths in the context
of second-order arithmetic, from being equivalent to to being
weaker than and incomparable with . In
particular, we identify a choice principle that, modulo induction,
lies strictly below the atomic model theorem principle and
implies the omitting partial types principle
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