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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
a distance function on Z^2
Posted: Oct 4, 2013 8:55 PM
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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.


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