495 research outputs found
Cross-intersecting families of vectors
Given a sequence of positive integers , let
denote the family of all sequences of positive integers
such that for all . Two families of sequences (or vectors),
, are said to be -cross-intersecting if no matter how we
select and , there are at least distinct indices
such that . We determine the maximum value of over all
pairs of - cross-intersecting families and characterize the extremal pairs
for , provided that . The case is
quite different. For this case, we have a conjecture, which we can verify under
additional assumptions. Our results generalize and strengthen several previous
results by Berge, Frankl, F\"uredi, Livingston, Moon, and Tokushige, and
answers a question of Zhang
Decomposition of multiple packings with subquadratic union complexity
Suppose is a positive integer and is a -fold packing of
the plane by infinitely many arc-connected compact sets, which means that every
point of the plane belongs to at most sets. Suppose there is a function
with the property that any members of determine
at most holes, which means that the complement of their union has at
most bounded connected components. We use tools from extremal graph
theory and the topological Helly theorem to prove that can be
decomposed into at most (-fold) packings, where is a constant
depending only on and .Comment: Small generalization of the main result, improvements in the proofs,
minor correction
Every graph admits an unambiguous bold drawing
Let r and w be fixed positive numbers, w < r. In a bold drawing of a graph, every vertex is represented by a disk of radius r, and every edge by a narrow rectangle of width w. We solve a problem of van Kreveld [10] by showing that every graph admits a bold drawing in which the region occupied by the union of the disks and rectangles representing the vertices and edges does not contain any disk of radius r other than the ones representing the vertices. © 2015, Brown University. All rights reserved
The visible perimeter of an arrangement of disks
Given a collection of n opaque unit disks in the plane, we want to find a
stacking order for them that maximizes their visible perimeter---the total
length of all pieces of their boundaries visible from above. We prove that if
the centers of the disks form a dense point set, i.e., the ratio of their
maximum to their minimum distance is O(n^1/2), then there is a stacking order
for which the visible perimeter is Omega(n^2/3). We also show that this bound
cannot be improved in the case of a sufficiently small n^1/2 by n^1/2 uniform
grid. On the other hand, if the set of centers is dense and the maximum
distance between them is small, then the visible perimeter is O(n^3/4) with
respect to any stacking order. This latter bound cannot be improved either.
Finally, we address the case where no more than c disks can have a point in
common. These results partially answer some questions of Cabello, Haverkort,
van Kreveld, and Speckmann.Comment: 12 pages, 5 figure
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