Re: Why Archimedes Plutonium will never proof the 4CT


Proginoskes wrote:
bill wrote:
Proginoskes wrote:
Gottfried Helms wrote:
Hi Christopher -

noting the effort, you've put into this discussion, I want
to add one remark, that at least to one reader you were able
to open eyes ;-)

Some years ago I had a similar discussion with some mathematician,
and I was in the position as AP: I could not understand, why
he denied my no-5-adjacent idea as a proof. Well, we had no
hassle about that, and I gave it a chance to possibly come out
to be understandable later. The current discussion solved that
inherent problem in my understanding. What I take now with me,
is that if the "non-5-adjacent"/"inside-triange-outside-triangle"
argument should be the core of the proof, then someone must prove
that any configuration can be reduced to a representation of
such (nested) non-5-adjacent-sub-configurations.
Is it that/something like, what A&H did with their computer-
proof?

It's like it. A&H's proof is direct, and they showed that there is a
fixed list of 1476 graphs such that: if you have a planar graph, then
it contains a graph on that list. (This is the "unavoidable
configuration" list.) AP's graphs do not form a list of unavoidable
configurations.

Every maximal planar graph contains a wheel graph of order six or less

(This is equivalent to the statement that some vertex has degree <= 5.)

Is this graph one of the 1475 A&H graphs?

The ones where the "hub" vertex has degree 1 -- 4. The wheels whose hub
vertex is 5 isn't one of them.

Evidently, A&H did not limit their analysis to maximal planar graphs.
Otherwise, they wouild have limited "hub" vertices to degrees >= 3.
Thomas et al limited "hub degrees to five or greater. Is that why they
have fewer configs. than A&H?

Bill

If you initially consider
only maximal planar graphs; then you have only to consider the wheel
graph of order 6.

Yes, because Kempe showed how to handle the other cases. Kempe thought
he handled the case where the wheel had order 6 (a hub vertex of degree
5), but he was wrong. A&H's 1475 graphs all contain a vertex of degree
5.

The decsion to study only maximal planar graphs is
justified by the observation that any planar graphs can be maximalized
by the addition of a sufficient number of edges.

That, and because if you add edges and get a coloring, that is also a
coloring of the original graph.

--- Christopher Heckman

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