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Plotting 3-trees in 6 vertices and evaluating their Prufer codes.

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  • Subject: [mg47612] Plotting 3-trees in 6 vertices and evaluating their Prufer codes.
  • From: "Diana" <diana53xiii at>
  • Date: Sun, 18 Apr 2004 04:15:09 -0400 (EDT)
  • Reply-to: "Diana" <diana53xiii at>
  • Sender: owner-wri-mathgroup at

Math friends,

I would appreciate help with the following problem.

I am trying to plot and evaluate Prufer codes for the 200 3-trees in 6
vertices. I will explain.

There are twenty triangles with 6 vertices. Assume that the vertices are
labeled a, b, c, d, e, f.

Take one triangle, (a, b, c). Now from vertex d, attach edges to a, b, and
c. Thus, you will have a tetrahedron in four vertices.

Now from vertex e, attach three edges to an existing triangle in the
tetrahedron. Call this new graph "5-gon". Then, with vertex f, attach it to
three edges of an existing triangle in "5-gon".

There are 200 of these 3-trees. I have gotten bogged down with redundancy,
plotting, and perhaps evaluating the Prufer codes with Mathematica.

The Prufer code is evaluated in the following way. Suppose that d and e are
adjacent to our triangle (a, b, c). Then, suppose f ties to triangle (a, b,

The Prufer code looks at the vertices of degree three in alphabetical order.

1. e is the first vertex of degree 3. It is adjacent to triangle (a, b, c).
Remove it.

2. Then, c is the first vertex of degree 3. c is adjacent to triangle (a, b,
d). Remove it.

3. You have a tetrahedron remaining, (a, b, d, f) The three latter letters
are (b, d, f) So, remove "a" leaving (b, d, f).

4. So, the Prufer code for this, first listing the vertex removed first, is
(a, b, c), (a, b, d), and (b, d, f).

There will be 200 triads of triples.

Can anyone help?



"God made the integers, all else is the work of man."
L. Kronecker, Jahresber. DMV 2, S. 19.

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