My proof of Four Colour Theorem

[Guest tutber]

Hi everybody!

 

As it seems to me I have found quite simple way to prove the "Four Colour Theorem". I am not a mathematician so I can't bear any illusions about my paper but I hope you can find it containing something interesting. You can see it at http://www.geocities.com/itutber.

 

Your comment would be very welcomed

 

Best wishes

Ilia Tutberidze

[matt grime]

The obvious flaw is that you do not show that your recolouring can be done in a consistent way without colouring two touching segments with the colour.

[Guest tutber]

Thank you Matt for your comment.

 

If I understand correctly, the flaw is that my "method" of map colouring is "limited" with coloring of two "touching segments" as you say.

 

Please can you explain why it is a flaw?

[matt grime]

The flaw is simple. the idea is that given some node where there are intetsections, that you can recolour the segments in a constistent fashionl the constistency you claim isn't in anyway proven to be consistent. it is an obvious observation, You might wish to prove tha your method shows that a some concrete diagram can be recoloured in 4 colours, HINT: you proof isn't valid in showing that nay arbitrary diagram can be recloured.

[Guest tutber]

Under arbitrary map I mean any map containing nodes with any arbitrary indices. You say, that my "proof isn't valid in showing that nay arbitrary diagram can be recloured". My question is following: Can you show me the other planar map which is out of "my definition" of arbitrary map? or, in other words: the map, which is not a particular case of the group of maps I have considered.

 

Of course, may be my Proof is not valid for the any arbitrary map even defined by me, or it is not consisitent, but it is the subject for other concrete discussion.

[matt grime]

your idea appears to be look at some node, recolour, proceed. However, you do not appear to show that this can be done in a consistent way that terminates with 4 colours. it is up to you to explain where I am wrong in this assesment, which i may be.

[Guest tutber]

I consider nodes just to show that in four-colour problem an arbitrary planar map may be transformed in such manner that the following cases take place.

1) We have only such nodes on the map, the index of which is K=3,

2) Each area represented on the map is 1-connected, and any closed line has at least two nodes.

By the way, afterwards I have rediscoveried that This is actually the standard approach to simplifying the discussion. There is a little book by Saaty and Kainen,

called "The Four-Color problem: Assaults and Conquest". and the statement above

amounts to what they call "Conjecture C'_2" on p.57.

That is all. In reality "my proof" begins after having this statement.

[matt grime]

At least that's one thing cleared up then. (I certainly have neither time, experience, nor inclination to give you a full assesment of it. Try asking in some place where the readership may contain a combinatorialist who cares about the 4-colour thorem. sci.math.research for instance. You almost certainly won't get any proper evaluation from this site.)