2,920 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
Quasi-Parallel Segments and Characterization of Unique Bichromatic Matchings
Given n red and n blue points in general position in the plane, it is
well-known that there is a perfect matching formed by non-crossing line
segments. We characterize the bichromatic point sets which admit exactly one
non-crossing matching. We give several geometric descriptions of such sets, and
find an O(nlogn) algorithm that checks whether a given bichromatic set has this
property.Comment: 31 pages, 24 figure
Contact graphs of ball packings
A contact graph of a packing of closed balls is a graph with balls as
vertices and pairs of tangent balls as edges. We prove that the average degree
of the contact graph of a packing of balls (with possibly different radii) in
is not greater than . We also find new upper bounds for
the average degree of contact graphs in and
Boxicity and separation dimension
A family of permutations of the vertices of a hypergraph is
called 'pairwise suitable' for if, for every pair of disjoint edges in ,
there exists a permutation in in which all the vertices in one
edge precede those in the other. The cardinality of a smallest such family of
permutations for is called the 'separation dimension' of and is denoted
by . Equivalently, is the smallest natural number so that
the vertices of can be embedded in such that any two
disjoint edges of can be separated by a hyperplane normal to one of the
axes. We show that the separation dimension of a hypergraph is equal to the
'boxicity' of the line graph of . This connection helps us in borrowing
results and techniques from the extensive literature on boxicity to study the
concept of separation dimension.Comment: This is the full version of a paper by the same name submitted to
WG-2014. Some results proved in this paper are also present in
arXiv:1212.6756. arXiv admin note: substantial text overlap with
arXiv:1212.675
Defective and Clustered Graph Colouring
Consider the following two ways to colour the vertices of a graph where the
requirement that adjacent vertices get distinct colours is relaxed. A colouring
has "defect" if each monochromatic component has maximum degree at most
. A colouring has "clustering" if each monochromatic component has at
most vertices. This paper surveys research on these types of colourings,
where the first priority is to minimise the number of colours, with small
defect or small clustering as a secondary goal. List colouring variants are
also considered. The following graph classes are studied: outerplanar graphs,
planar graphs, graphs embeddable in surfaces, graphs with given maximum degree,
graphs with given maximum average degree, graphs excluding a given subgraph,
graphs with linear crossing number, linklessly or knotlessly embeddable graphs,
graphs with given Colin de Verdi\`ere parameter, graphs with given
circumference, graphs excluding a fixed graph as an immersion, graphs with
given thickness, graphs with given stack- or queue-number, graphs excluding
as a minor, graphs excluding as a minor, and graphs excluding
an arbitrary graph as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in
the Electronic Journal of Combinatoric
HipergrĂĄfok = Hypergraphs
A projekt cĂ©lkitƱzĂ©seit sikerĂŒlt megvalĂłsĂtani. A nĂ©gy Ă©v sorĂĄn több mint szĂĄz kivĂĄlĂł eredmĂ©ny szĂŒletett, amibĆl eddig 84 dolgozat jelent meg a tĂ©ma legkivĂĄlĂłbb folyĂłirataiban, mint Combinatorica, Journal of Combinatorial Theory, Journal of Graph Theory, Random Graphs and Structures, stb. SzĂĄmos rĂ©gĂłta fennĂĄllĂł sejtĂ©st bebizonyĂtottunk, egĂ©sz rĂ©gi nyitott problĂ©mĂĄt megoldottunk hipergrĂĄfokkal kapcsolatban illetve kapcsolĂłdĂł terĂŒleteken. A problĂ©mĂĄk nĂ©melyike sok Ă©ve, olykor több Ă©vtizede nyitott volt. Nem egy közvetlen kutatĂĄsi eredmĂ©ny, de szintĂ©n bizonyos Ă©rtĂ©kmĂ©rĆ, hogy a rĂ©sztvevĆk egyike a NorvĂ©g KirĂĄlyi AkadĂ©mia tagja lett Ă©s elnyerte a Steele dĂjat. | We managed to reach the goals of the project. We achieved more than one hundred excellent results, 84 of them appeared already in the most prestigious journals of the subject, like Combinatorica, Journal of Combinatorial Theory, Journal of Graph Theory, Random Graphs and Structures, etc. We proved several long standing conjectures, solved quite old open problems in the area of hypergraphs and related subjects. Some of the problems were open for many years, sometimes for decades. It is not a direct research result but kind of an evaluation too that a member of the team became a member of the Norvegian Royal Academy and won Steele Prize
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