640 research outputs found
Isoperimetric Inequalities on Hexagonal Grids
We consider the edge- and vertex-isoperimetric probem on finite and infinite
hexagonal grids: For a subset W of the hexagonal grid of given cardinality, we
give a lower bound for the number of edges between W and its complement, and
lower bounds for the number of vertices in the neighborhood of W and for the
number of vertices in the boundary of W. For the infinite hexagonal grid the
given bounds are tight
The isoperimetric constant of the random graph process
The isoperimetric constant of a graph on vertices, , is the
minimum of , taken over all nonempty subsets
of size at most , where denotes the set of
edges with precisely one end in . A random graph process on vertices,
, is a sequence of graphs, where
is the edgeless graph on vertices, and
is the result of adding an edge to ,
uniformly distributed over all the missing edges. We show that in almost every
graph process equals the minimal degree of
as long as the minimal degree is . Furthermore,
we show that this result is essentially best possible, by demonstrating that
along the period in which the minimum degree is typically , the
ratio between the isoperimetric constant and the minimum degree falls from 1 to
1/2, its final value
Vertex Isoperimetric Inequalities for a Family of Graphs on Z^k
We consider the family of graphs whose vertex set is Z^k where two vertices
are connected by an edge when their l\infty-distance is 1. We prove the optimal
vertex isoperimetric inequality for this family of graphs. That is, given a
positive integer n, we find a set A \subset Z^k of size n such that the number
of vertices who share an edge with some vertex in A is minimized. These sets of
minimal boundary are nested, and the proof uses the technique of compression.
We also show a method of calculating the vertex boundary for certain subsets
in this family of graphs. This calculation and the isoperimetric inequality
allow us to indirectly find the sets which minimize the function calculating
the boundary.Comment: 19 pages, 2 figure
Optimal Random Matchings, Tours, and Spanning Trees in Hierarchically Separated Trees
We derive tight bounds on the expected weights of several combinatorial
optimization problems for random point sets of size distributed among the
leaves of a balanced hierarchically separated tree. We consider {\it
monochromatic} and {\it bichromatic} versions of the minimum matching, minimum
spanning tree, and traveling salesman problems. We also present tight
concentration results for the monochromatic problems.Comment: 24 pages, to appear in TC
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