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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.