- #1
fenstip
- 10
- 0
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, now, the colours (or after re-colour) on the nodes of the map are 1,2,3,4 or less. The map at the above showing just the nodes at the very edge (rim? outside?) of the map (there are still other patterns to consider), then if I could change the colour on some of them in my way, and put back A and B, the map is four colours. I put some Kempe Chain to the current nodes which are at the edge, then I'm not going to change their colours because it's difficult. But the Kempe Chain blocks other connections, that's the point I can use.
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 edge are CghiDEF (other nodes are existed in the map but I didn't have them shown due to convenience). Remove AB from the map, call that map 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 Kempe Chain block C,h to genarate 1-3 an 1-4 Kempe 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. Now, put back AB, map is four colours. So the best way to keep the asumption is to set B.d 7 or above. Now, do the same thing to DEF.
Sorry if my English has you confused.
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, now, the colours (or after re-colour) on the nodes of the map are 1,2,3,4 or less. The map at the above showing just the nodes at the very edge (rim? outside?) of the map (there are still other patterns to consider), then if I could change the colour on some of them in my way, and put back A and B, the map is four colours. I put some Kempe Chain to the current nodes which are at the edge, then I'm not going to change their colours because it's difficult. But the Kempe Chain blocks other connections, that's the point I can use.
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 edge are CghiDEF (other nodes are existed in the map but I didn't have them shown due to convenience). Remove AB from the map, call that map 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 Kempe Chain block C,h to genarate 1-3 an 1-4 Kempe 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. Now, put back AB, map is four colours. So the best way to keep the asumption is to set B.d 7 or above. Now, do the same thing to DEF.
Sorry if my English has you confused.