259 research outputs found
Cubic Partial Cubes from Simplicial Arrangements
We show how to construct a cubic partial cube from any simplicial arrangement
of lines or pseudolines in the projective plane. As a consequence, we find nine
new infinite families of cubic partial cubes as well as many sporadic examples.Comment: 11 pages, 10 figure
Dimension reduction by random hyperplane tessellations
Given a subset K of the unit Euclidean sphere, we estimate the minimal number
m = m(K) of hyperplanes that generate a uniform tessellation of K, in the sense
that the fraction of the hyperplanes separating any pair x, y in K is nearly
proportional to the Euclidean distance between x and y. Random hyperplanes
prove to be almost ideal for this problem; they achieve the almost optimal
bound m = O(w(K)^2) where w(K) is the Gaussian mean width of K. Using the map
that sends x in K to the sign vector with respect to the hyperplanes, we
conclude that every bounded subset K of R^n embeds into the Hamming cube {-1,
1}^m with a small distortion in the Gromov-Haussdorf metric. Since for many
sets K one has m = m(K) << n, this yields a new discrete mechanism of dimension
reduction for sets in Euclidean spaces.Comment: 17 pages, 3 figures, minor update
Graphs that are isometrically embeddable in hypercubes
A connected 3-valent plane graph, whose faces are - or 6-gons only, is
called a {\em graph }. We classify all graphs , which are isometric
subgraphs of a -hypercube .Comment: 18 pages, 25 drawing
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