27 research outputs found
Geometric Spanning Cycles in Bichromatic Point Sets
Given a set of points in the plane each colored either red or blue, we find
non-self-intersecting geometric spanning cycles of the red points and of the
blue points such that each edge of the red spanning cycle is crossed at most
three times by the blue spanning cycle and vice-versa
The hamburger theorem
We generalize the ham sandwich theorem to measures in as
follows. Let be absolutely continuous finite
Borel measures on . Let for , and assume that . Assume that for every . Then there
exists a hyperplane such that each open halfspace defined by
satisfies for every
and . As a
consequence we obtain that every -colored set of points in
such that no color is used for more than points can be
partitioned into disjoint rainbow -dimensional simplices.Comment: 11 pages, 2 figures; a new proof of Theorem 8, extended concluding
remark
-covering red and blue points in the plane
We say that a finite set of red and blue points in the plane in general
position can be -covered if the set can be partitioned into subsets of
size , with points of one color and point of the other color, in
such a way that, if at each subset the fourth point is connected by
straight-line segments to the same-colored points, then the resulting set of
all segments has no crossings. We consider the following problem: Given a set
of red points and a set of blue points in the plane in general
position, how many points of can be -covered? and we prove
the following results:
(1) If and , for some non-negative integers and ,
then there are point sets , like -equitable sets (i.e.,
or ) and linearly separable sets, that can be -covered.
(2) If , and the points in are in convex position,
then at least points can be -covered, and this bound is tight.
(3) There are arbitrarily large point sets in general position,
with , such that at most points can be -covered.
(4) If , then at least points of
can be -covered. For , there are too many red points and at
least of them will remain uncovered in any -covering.
Furthermore, in all the cases we provide efficient algorithms to compute the
corresponding coverings.Comment: 29 pages, 10 figures, 1 tabl
Rainbow polygons for colored point sets in the plane
Given a colored point set in the plane, a perfect rainbow polygon is a simple
polygon that contains exactly one point of each color, either in its interior
or on its boundary. Let denote the smallest size
of a perfect rainbow polygon for a colored point set , and let
be the maximum of
over all -colored point sets in general position; that is, every -colored
point set has a perfect rainbow polygon with at most
vertices. In this paper, we determine the values
of up to , which is the first case where
, and we prove that for , Furthermore, for a -colored set of points in the plane in general
position, a perfect rainbow polygon with at most vertices can be computed in time.Comment: 23 pages, 11 figures, to appear at Discrete Mathematic
Rainbow polygons for colored point sets in the plane
Given a colored point set in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color, either in its interior or on its boundary. Let rb-index(S) denote the smallest size of a perfect rainbow polygon for a colored point set S, and let rb-index(k) be the maximum of rb-index(S) over all k-colored point sets in general position; that is, every k-colored point set S has a perfect rainbow polygon with at most rb-index(k) vertices. In this paper, we determine the values of rb-index(k) up to k=7, which is the first case where rb-index(k)¿k, and we prove that for k=5, [Formula presented] Furthermore, for a k-colored set of n points in the plane in general position, a perfect rainbow polygon with at most [Formula presented] vertices can be computed in O(nlogn) time. © 2021 Elsevier B.V
K1,3-covering red and blue points in the plane
We say that a finite set of red and blue points in the plane in general position can be K1, 3-covered if the set can be partitioned into subsets of size 4, with 3 points of one color and 1 point of the other color, in such a way that, if at each subset the fourth point is connected by straight-line segments to the same-colored points, then the resulting set of all segments has no crossings. We consider the following problem: Given a set R of r red points and a set B of b blue points in the plane in general position, how many points of R ¿ B can be K1, 3-covered? and we prove the following results: (1) If r = 3g + h and b = 3h + g, for some non-negative integers g and h, then there are point sets R ¿ B, like {1, 3}-equitable sets (i.e., r = 3b or b = 3r) and linearly separable sets, that can be K1, 3-covered. (2) If r = 3g + h, b = 3h + g and the points in R ¿ B are in convex position, then at least r + b - 4 points can be K1, 3-covered, and this bound is tight. (3) There are arbitrarily large point sets R ¿ B in general position, with r = b + 1, such that at most r + b - 5 points can be K1, 3-covered. (4) If b = r = 3b, then at least 9 8 (r + b- 8) points of R ¿ B can be K1, 3-covered. For r > 3b, there are too many red points and at least r - 3b of them will remain uncovered in any K1, 3-covering. Furthermore, in all the cases we provide efficient algorithms to compute the corresponding coverings
Recommended from our members
Discrete Geometry
A number of important recent developments in various branches of discrete geometry were presented at the workshop. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics or algebraic geometry. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry