> < ^ Date: Thu, 20 Nov 2003 09:40:57 -0500
< ^ From: Drew Krause <drkrause@mindspring.com >
> ^ Subject: pairwise nonintersection graphs?

I would like to be able to do the following using GAP and the grape
package. I need help in the proper way to formulate the problem in terms
of the software:

(1) Take the set of non-identical pairs (x,y) from a set of n elements.
Two pairs would be connected if the intersection of its members was
empty. For example. the pairs (1,3) and (4,2) would be connected, but
(1,3) and (2,3) would not.

(2) Generalize the above to tuples of length greater than 2. For
example, (1,3,4) and (2, 5, 6) would be connected since there is no
intersection among the members.

(3) Generalize to include m additional settings of each tuple-member,
such that when the conditions of (1) and (2) are satisfied above, they
will be satisfied for any setting of each tuplet member. For example, if
(1,3) and (2,4) are connected, then the pairs ((1,a),(3,b)) and
((2,c),(4,d)) are also connected for a,b,c,d in a new domain.

Forgive the inelegant formulation above -- my understanding of group &
graph theory far from complete. I have been using GAP for several years
to solve combinatoric and other problems related to my work in music
composition. In this case, briefly, I want to construct graphs that
connect dyads and other chords which do not contain movement.by unison
(cases 1 or 2) or octave (case 3). Any help will be greatly appreciated.

Drew Krause
New York, NY

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