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