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In the paper On the geographical problem of the Four Colors by Kempe, at the end of page 194, it is stated that

It will readily be seen that we can interchange the colours of the districts in one or miore of the red and green regions without doing so in any others, and the map will still be properly coloured. The same remarks apply to the regions composed of districts of any other pair of colour. Now if a region composed of districts of any pair of colours, say red and green as before, be of either of the forms shown in Figures 3 and 4, it will separate the surface into twvo parts, so that we map be quite certain that no yellow or blue districts in one part can belong, to the same yellow and blue region as any yellow or blue district in the other part

enter image description here

But I'm having trouble understading his argument. I mean as far as I understood, those lines in the figures are the boundaries of regions which contain districs, and if we change the colors of district, say blue and green, in one region only, there will be no problem, but can't a district of blue color in the inner region have a neighbour with green color in the outer region ?

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    $\begingroup$ I'm not sure why you think we'll take the time to think about that quote if you don't even take the time to proofread it to make it understandable. $\endgroup$
    – joriki
    Jun 23 '18 at 4:09
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    $\begingroup$ I haven't read that paper in a long time, but as I recall, it was very difficult to follow the argument in terms of maps. That is, I think, why an error that jumps off the page at you if stated in terms of graphs, went unnoticed for ten years when stated in terms of maps. $\endgroup$
    – saulspatz
    Jun 23 '18 at 4:21
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    $\begingroup$ @joriki Because nobody (including me) answers question because they want to "help" people; This is not a charity. If the question interesting for you, you take to challenge to answer, if not, you do not have to. I have put the link of the article into the question, for in case situations also. Plus, I have proofread the quote, but right now, I'm still trying to understand the argument and the article, so I'm not sorry if the quote contains a typo like "twvo", and you do not answer to the question just because of that. $\endgroup$
    – Our
    Jun 23 '18 at 4:40
  • $\begingroup$ Of course, the title of the paper is On the Geographical Problem of the Four Colours. $\endgroup$ Jun 23 '18 at 4:55
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    $\begingroup$ It contains more serious typos than that; it says "so that we map be" and dozens (currently $20$) people have to stop and figure out what that means instead of one person (you) taking a second to check it against the text. I do sometimes answer questions to help people, and I don't see a reason to put scare quotes around that word. Of course, if you're not interested in such people answering your questions, that's OK. Perhaps you could indicate it at the top in order not to waste our time. (It would still be efficient and kind towards the non-charitable non-helpers if you proofread the text.) $\endgroup$
    – joriki
    Jun 23 '18 at 5:49
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enter image description here

I couldn't figure out how to post an image in a comment, so ...

The map has two red/green regions, A and D. It has two blue/yellow regions, B and C. Region B consists of a single blue district. If green and red were switched in region A, the other regions would be unaffected. Every district in A is adjacent to other districts in A or blue/yellow districts in B or C.

The single district in region B can be blue or yellow.

If we wanted blue/green regions and red/yellow regions instead, there would be a single blue/green region and 4 red/yellow regions.

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