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6 votes

Estimating shortest paths in planar drawings of graphs

Looks like Saúl RM has already achieved an arbitrarily high distance. Here's a simpler construction that at least gets distance $> 1$. The square has vertices $(\pm1/2, \pm1/2)$. The vertices near ...
Noam D. Elkies's user avatar
5 votes
Accepted

Estimating shortest paths in planar drawings of graphs

Here are triangulations of a side $1$ square with vertices at a arbitrarily high distance of all the four vertices of the square. The sides of the side $1$ square are not edges but it is easy to see ...
Saúl RM's user avatar
  • 8,741
9 votes
Accepted

existence of triangulations of manifolds

In general, the answer is no. For an example, suppose that $K$ is the boundary of the four-simplex. Thus $K$ is a triangulation of the three-sphere. Now every closed connected oriented surface of ...
Sam Nead's user avatar
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