# The four color map?

## Main Question or Discussion Point

[SOLVED] The four color map??

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

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 any one 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 isnt important then they may as well be points on a circle and so be very symetrical, so the order wont 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 dont 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

HallsofIvy
Homework Helper
I don't know why it wouldn't be called a proof. Appel and Haken proved that in 1976.

mgb_phys
Homework Helper
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.

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.

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!!

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!!]

HallsofIvy
Homework Helper
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".

CRGreathouse