65 research outputs found
Colorful Strips
Given a planar point set and an integer , we wish to color the points with
colors so that any axis-aligned strip containing enough points contains all
colors. The goal is to bound the necessary size of such a strip, as a function
of . We show that if the strip size is at least , such a coloring
can always be found. We prove that the size of the strip is also bounded in any
fixed number of dimensions. In contrast to the planar case, we show that
deciding whether a 3D point set can be 2-colored so that any strip containing
at least three points contains both colors is NP-complete.
We also consider the problem of coloring a given set of axis-aligned strips,
so that any sufficiently covered point in the plane is covered by colors.
We show that in dimensions the required coverage is at most .
Lower bounds are given for the two problems. This complements recent
impossibility results on decomposition of strip coverings with arbitrary
orientations. Finally, we study a variant where strips are replaced by wedges
Hitting sets and colorings of hypergraphs
In this paper we study the minimal size of edges in hypergraph families which
guarantees the existence of a polychromatic coloring, that is, a -coloring
of a vertex set such that every hyperedge contains a vertex of all color
classes. We also investigate the connection of this problem with -shallow
hitting sets: sets of vertices that intersect each hyperedge in at least one
and at most vertices.
We determine in some hypergraph families the minimal for which a
-shallow hitting set exists.
We also study this problem for a special hypergraph family, which is induced
by arithmetic progressions with a difference from a given set. We show
connections between some geometric hypergraph families and the latter, and
prove relations between the set of differences and polychromatic colorability
Proper Coloring of Geometric Hypergraphs
We study whether for a given planar family F there is an m such that any finite set of points can be 3-colored such that any member of F that contains at least m points contains two points with different colors. We conjecture that if F is a family of pseudo-disks, then m=3 is sufficient. We prove that when F is the family of all homothetic copies of a given convex polygon, then such an m exists. We also study the problem in higher dimensions
Generalisation : graphs and colourings
The interaction between practice and theory in mathematics is a central theme. Many mathematical structures and theories result from the formalisation of a real problem. Graph Theory is rich with such examples. The graph structure itself was formalised by Leonard Euler in the quest to solve the problem of the Bridges of Königsberg. Once a structure is formalised, and results are proven, the mathematician seeks to generalise. This can be considered as one of the main praxis in mathematics. The idea of generalisation will be illustrated through graph colouring. This idea also results from a classic problem, in which it was well known by topographers that four colours suffice to colour any map such that no countries sharing a border receive the same colour. The proof of this theorem eluded mathematicians for centuries and was proven in 1976. Generalisation of graphs to hypergraphs, and variations on the colouring theme will be discussed, as well as applications in other disciplines.peer-reviewe
Facial Achromatic Number of Triangulations with Given Guarding Number
A (not necessarily proper) -coloring of a graph on a surface is a {\em facial -complete -coloring} if every -tuple of colors appears on the boundary of some face of . The maximum number such that has a facial -complete -coloring is called a {\em facial -achromatic number} of , denoted by . In this paper, we investigate the relation between the facial 3-achromatic number and guarding number of triangulations on a surface, where a {\em guarding number} of a graph embedded on a surface, denoted by \gd(G), is the smallest size of its {\em guarding set} which is a generalized concept of guards in the art gallery problem. We show that for any graph embedded on a surface, \psi_{\Delta(G^*)}(G) \leq \gd(G) + \Delta(G^*) - 1, where is the largest face size of . Furthermore, we investigate sufficient conditions for a triangulation on a surface to satisfy \psi_{3}(G) = \gd(G) + 2. In particular, we prove that every triangulation on the sphere with \gd(G) = 2 satisfies the above equality and that for one with guarding number , it also satisfies the above equality with sufficiently large number of vertices
Proper coloring of geometric hypergraphs
We study whether for a given planar family F there is an m such that any finite set of points can be 3-colored such that any member of F that contains at least m points contains two points with different colors. We conjecture that if F is a family of pseudo-disks, then m = 3 is sufficient. We prove that when F is the family of all homothetic copies of a given convex polygon, then such an m exists. We also study the problem in higher dimensions. © Balázs Keszegh and Dömötör Pálvölgyi
An abstract approach to polychromatic coloring: shallow hitting sets in ABA-free hypergraphs and pseudohalfplanes
The goal of this paper is to give a new, abstract approach to
cover-decomposition and polychromatic colorings using hypergraphs on ordered
vertex sets. We introduce an abstract version of a framework by Smorodinsky and
Yuditsky, used for polychromatic coloring halfplanes, and apply it to so-called
ABA-free hypergraphs, which are a generalization of interval graphs. Using our
methods, we prove that (2k-1)-uniform ABA-free hypergraphs have a polychromatic
k-coloring, a problem posed by the second author. We also prove the same for
hypergraphs defined on a point set by pseudohalfplanes. These results are best
possible. We could only prove slightly weaker results for dual hypergraphs
defined by pseudohalfplanes, and for hypergraphs defined by pseudohemispheres.
We also introduce another new notion that seems to be important for
investigating polychromatic colorings and epsilon-nets, shallow hitting sets.
We show that all the above hypergraphs have shallow hitting sets, if their
hyperedges are containment-free
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