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Topic: a distance function on Z^2
Replies: 1   Last Post: Oct 6, 2013 5:23 PM

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Posts: 12,067
Registered: 7/15/05
Re: a distance function on Z^2
Posted: Oct 6, 2013 5:23 PM
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quasi wrote:
>For P,Q in R^2, let |P - Q| denote the usual Euclidean distance
>from P to Q.
>Define a graph G with vertex set Z^2 such that, for distinct
>points P,Q in Z^2, PQ is an edge iff
> (1) |P - Q| is a positive integer.
> (2) The line segment PQ is not horizontal or vertical.
>For P,Q in Z^2, let d(P,Q) denote the graph-theoretic distance
>from P to Q in the graph G.
>I'll pose 3 problems ...
>Problem (1) is fairly easy.
>Problem (2) is medium hard.
>For problem (3), I don't yet have a solution.
>(1) Show that G is connected.
>(2) Find points P,Q in Z^2 such that d(P,Q) = 3.
>(3) Prove or disprove: The diameter of G is 3.

I can now prove (3).

It's not hard.


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