Date: Feb 26, 2013 3:03 PM
Author: Ken.Pledger@vuw.ac.nz
Subject: Re: A topological problem

In article <ap4910Fk250U1@mid.dfncis.de>, <fc3a501@uni-hamburg.de> 
wrote:

> Warning, incoming lousy ASCII, change to fixed font :-)
>
> o o o
> | | |
> | O |
> \ /|\ /
> X | X
> / \|/ \
> | O |
> | | |
> o o o
>
> The lines |/\ are tied to the unmovable nodes Oo.
> (As you see, four lines come out of O and one out of o.)
> X denotes a crossing, which is like a virtual crossing
> from knot theory, i.e. you can move it ad lib and any
> line over any other. Should they cross in the process,
> well duh, then you have more crossings.)
>
> Can you move the lines around such that no horizontal
> line going through this graph cuts more than three
> of these lines? I think no, but can you lend me a
> formal proof?



Can you move a crossing past an o? If so, the thing is a planar
graph: just move the four corner o's, top to bottom and bottom to top.

o o o
\|/
O
|
O
/|\
o o o

That fulfils your condition about horizontal lines. But is there some
other restriction on moving the crossings?

Ken Pledger.