Four Color Map Theorem: Proving the Impossibility?

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Discussion Overview

The discussion revolves around the Four Color Map Theorem, specifically exploring the impossibility of creating a five-color map and the nature of proofs in this context. Participants engage in reasoning about the relationships between regions on maps and the implications for coloring them, touching on both theoretical and practical aspects of the theorem.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant describes their reasoning process for concluding that a five-color map cannot exist, based on the inability to connect all regions without crossing lines.
  • Another participant references the proof by Appel and Haken from 1976, suggesting it established the theorem's validity.
  • Some participants argue that trying various cases does not constitute a proof, especially when there are infinitely many cases to consider.
  • It is noted that there are maps that cannot be colored with four colors, which do not contain four mutually adjoining regions, challenging the initial reasoning presented.
  • One participant expresses curiosity about specific maps that illustrate the need for four colors, mentioning Nevada as an example surrounded by five states.
  • There is a correction regarding the interpretation of statements about maps that can or cannot be colored with three or four colors, indicating some confusion in the discussion.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the nature of proof regarding the five-color map and the validity of the initial reasoning. Multiple competing views remain regarding what constitutes a proof and the implications of the Four Color Map Theorem.

Contextual Notes

Participants highlight limitations in reasoning, such as the dependence on specific configurations of regions and the complexity of proving general cases versus specific examples.

phlegmy
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[SOLVED] The four color map??

so i guess most of you know about the four color map theorem.

i read about it in a book a couple of days ago and had some idle brain time today while driving a tractor.
i scribbled out some maps on the dirt on the windows and always seems to enclose one region of the map in my attempt to find a five color map.

i then reasoned that if a five color map existed then there must be a section of it containing 5 mutually adjoining regions. and that if i could stand in anyone of the regions i should be able to walk to any of the other four without crossing another. and so i thought.. i'll draw dots to represent regions and lines to represent the path i'd take from one to another. so in joining the dots i again come to a stop, it is impossible for me to join all dots to all dots without crossing lines, i.e. distroying boarders between regions. so i thought, i'll draw another five dots and this time be carefull not to cross lines, but i end up in a situation where in order to connect two dots i must enclose a "deficient" dot.

so i figure there can never exist 5 mutally adjoining regions unless i can connect all the dots without crossing lines. if there and only 4 mutally adjoining regions [which can be easily done with dots and lines] i only need 4 colours.

i then wonder, does the orientations of the dots matter?, and i think, no, because i can alter my path,
i then wonder does it matter what order i connect the dots in, and i think, perhaps, but if the orientation of the dots isn't important then they may as well be points on a circle and so be very symetrical, so the order won't be too important

then i feel very happy that i will no longer waste my time trying to draw a five color map on the side of a trailer or the inside of a tractor cab. I've satisfied myself it cannot be done. then i wonder,.. does that consititute a proof.

so i do some googleing and find that some equally clever people have already done pretty much the same thing. but they don't call it proof??

so I'm figureing, I'm satisfied i cannot connect all the 5 dots, but i havnt "prooved" it. it's pretty obvious to someone who tries they cannot do it. [much more obvious than trying to draw funny maps!) so why is it not accepted as a proof?? or has anyone any thougths on this

apologies as always for my spelling
 
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I don't know why it wouldn't be called a proof. Appel and Haken proved that in 1976.
 
Trying as many cases as you can think of isn't a proof - if there is an infinite number of cases to try.
What I remember being proved is that all maps can be split it one of a large (but finite) number of examples and then they 'proved' on a computer that all of these maps were 4 colour.
 
phlegmy said:
i then reasoned that if a five color map existed then there must be a section of it containing 5 mutually adjoining regions.

This is not enough for a proof. There are maps that you can't colour with 4 colours, that do not contain 4 mutually adjoining regions.
 
kamerling said:
This is not enough for a proof. There are maps that you can't colour with 4 colours, that do not contain 4 mutually adjoining regions.

thanks guys i suppose that would explain why its not a solution!
i'd be interested to see the map you refer to!
 
phlegmy said:
thanks guys i suppose that would explain why its not a solution!
i'd be interested to see the map you refer to!

I'm sure you must have seen a map like that many times. Any map with an area that is surrounded by a ring of an odd number of areas needs four colours. For example Nevada and the five states surrounding it.
 
i see.. I've just done it!
interesting how adding, or removing one region will reduce it to a 3 color map!
[never seen a map of nevada!]
 
kamerling said:
This is not enough for a proof. There are maps that you can't colour with 4 colours, that do not contain 4 mutually adjoining regions.

I think you mean "There are maps that you can colour with 4 colours".
 
HallsofIvy said:
kamerling said:
This is not enough for a proof. There are maps that you can't colour with 4 colours, that do not contain 4 mutually adjoining regions.
I think you mean "There are maps that you can colour with 4 colours".

I thought kamerling meant, "There are maps you can't color with 3 colors.".
 

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