A 4 colours thoerem

[traevkuq]

line-node-map is formed by lines and nodes, commonly used by the one (including me) who reseaches four colour theorem.

Smallest five colours map is the one with fewest number of nodes and couldn't be coloured less than five colours, and also, it can be a complete map by adding lines not nodes. I remove temporarily two adjacent nodes (A and B) from the map, the colours on the nodes of the rest map are 1,2,3,4 or less. the map at the down side showing just the nodes at the very edge of the map (not all possibilities), then If I could change the colour on someone of them, and put back A and B, the map is four colours. So I put some Kempers Chain between them, then I'm not going to change their colours because it's difficult, so I will keep the assumption correct. But the Kemper Chain blocks other connections, that's the point I can use.

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Assuming A.c=5 (colour of A), A.d=5 (degree(s) or how many node(s) connecting to), B.d=6, after removing AB, the nodes at the very edge are CghiDEF (other nodes are in the map but I didn't have them on). Remove AB from the map, call that G2. Supposed C.c=1, g.c=2, h.c=1, i.c=3, D.c=1, E.c=3, F.c=2, if there is no Kempe chain between g and F, I will change g.c to 4, causing B.d=6 impossible. But to consider fully, I have to set a Kempe chain between g and F, the same reason, Kempe chain between gi, iE, just an example. After that the Kemper Chain block C,h to genarate 1-3 an 1-4 Kemper Chain, so I change C.c to 3, (it may generate 3-2 between between i, and so on but this couln't stop colour change ), change D.c to 3.

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I will keep the number of examples or patterns less by my idea. Erase all colours on the map, pull and drag C,h together by extending the lines to C,h, that will merge C,h to form a node called V1, the same reason, merge V1 and D to form V2, right now there are two nodes less, so after putting colours on the map, can be only four or less colour. Splitting the V1 node from inside of the map to outside, all colours at the very outside of the map maybe four, but good news is 3 nodes (ChD) have the same colour.

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Also, I can do the same thing to other patterns (A.d=5,B.d=2), (A.d=6,B.d=2), And some of the Kemper Chains have the different end points, lengths, breaking throughs (The Kempe Chain can go through other kempe chain, like Kempe Chain between gi goes through Kempe Chain between iE). It's very easy to get rid of these situations. If I'm wrong, you are easy to tell like P.J.Heawood because I heavily use Kempe Chain. (Here would insert an image but cannot add another attachment, 3 is the maximum)

If the above proof is right, then if A.d is 5, then the nodes.d connected to A is equal to or more than 7, if A.d=6, then the nodes.d connected to A is equal to or more than 6. if A.d>=7, then the nodes.d connected to A is equal to or more than 5. The node colouring 5 can be anyone in the map.

Now you have got some information of the smallest five colours map, which will help me get rid of 4c theorem.

Later proof, I will not focus on the colours much, but whether the smallest five colours map can match the degrees I have proved. If not, the smallest five colours map doesn't exist.

It's pity for me that I can't speak more english in the later proof.

[traevkuq]

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[traevkuq]

The V-merge may make the outside nodes the forth color, and couldn't merge Ch if Ch adjacent. Now I have better and easier proof to replace it.

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[traevkuq]

Some words about V-merge should be removed .

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[traevkuq]

Around the Internet, there are some other easy proofs on four color theorem which is wrong in the same way. The author provides some simple patterns, which all can be colored within five, and merge them together, the new map can be four color too, and some countries are isolated. Then the author says if he continue adding patterns to the map, four color is enough to color the map.

After merging the patterns, may be the color on several nodes have to change but may loop back to the the original place (node), or the existing color force the color 5 appeared, how to keep the colors less than 5 is not described in the past patterns, so the color change fail.

[traevkuq]

Around the Internet, there are some other easy proofs on four color theorem which is wrong in the same way. The author provides some simple patterns, which all can be colored within five, and merge them together, the new map can be four color too, and some countries are isolated. Then the author says if he continue adding patterns to the map, four color is enough to color the map.

After merging the patterns, may be the color on several nodes have to change but may loop back to the the original place (node), or the existing color force the color 5 appeared, how to keep the colors less than 5 is not described in the past patterns, so the color change fail.

But one of the necessary properties of the assuming map is degree, which can't be satisfied by any map.

[traevkuq]

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[traevkuq]

Maybe not only one smallest 5 color map, which has different nodes or connections. So the smallest 5 color map in my proof is just one case, while other smallest 5 color maps shared the same proof.

AI translator is much better at translation than me, I'm going to put the new translation of my text on.

[traevkuq]

It's not necessary to replace V-merge, that if I place B to the original place, ChB circles nodes if Ch are connected. Similar to Dh. makes ChD same color.

[traevkuq]

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