Talk:Four color theorem/Archive 3

message: :The four colour theorem does not arise out of and has no origin in practical cartography. According to Kenneth May, a mathematical historian who studied a sample of atlases in the Library of Congress, there is no tendency to minimise the number of colors used. Maps utilizing only four colors are rare, and those that do usually require only three.
Can someone check the last sentence in this paragraph? It is wrong (maps utilizing four colors use only three colors?) . I don't know anything about maps but I guess it should be:
Maps utilizing only four colors are rare, and those that MINIMIZE THE NUMBER OF COLORS USED do usually require only three.  But then it seems strange to me that cartographic maps that minimize colors usually need three colors.
:I reread this, I misunderstood it. They're saying that most maps use more than four colours, and when four colours are used, usually three suffice. In other words, mapmakers tend to pretty much always use more colours than necessary. Not sure how to rephrase this. Deco 1 July 2005 00:19 (UTC)
::I actually think it's pretty clear as it stands. Dmharvey  Talk 1 July 2005 00:51 (UTC)
:On another note, I'd think that the requirement on some practical maps that non-contiguous regions have the same color (for example: coloring Alaska , Hawaii and the continental US all the same color on a world map) would fuck up the theorem, however I'm not sure if this is true or not. Anyone know? It should be in the "Not for map makers" section if it is true. -- Tyler 09:22, 27 September 2005 (UTC)
replies: text: *That's a good point.  While the US is not a good example (I don't think that there are any on Earth anymore, but I might be wrong) the idea destroys the theorem for actual maps with non-contiguous regions of the same color. --Apyule 13:26, 27 September 2005 (UTC)
replies: text:  Isn't this already mentioned in the opening paragraph? Dmharvey  Talk 23:01, 27 September 2005 (UTC)
replies: text:  And in the whole section at the bottom ("real-world counterexamples")! I have no idea how I missed all that... I suppose I expected to find it in the section "Not for map-makers" and assumed it wasn't there at all. D'oh! -- Tyler 23:41, 27 September 2005 (UTC)

text: * if there is only one discontinuous country then 7 colors neccesarily suffice: if you imagine an underground tunnel connecting the two areas and chonsider the walls of the tunnel, you get a torus surface. And as the article says, 7 colors suffice to color any torus.Honnza 08:39, 14 July 2006 (UTC)
thread_title:  Not for map-makers 

message: Some of the material added by Ewlyahoocom doesn't seem particularly encyclopaedic. Anyone agree? Dmharvey  Talk 21:09, 29 September 2005 (UTC)
replies: text: I agree. Ewlyahoocom 13:50, 30 September 2005 (UTC)
replies: text:  I've looked around a little bit. How-to suggests Wikibooks as a better place to put the How-To kind of information I'm envisioning. I'll investigate some more. Ewlyahoocom 08:21, 2 October 2005 (UTC)
thread_title:  Comment on recent edits 

message: This theorum looks like:
No matter how you divide a 2D plane, you can't construct a series of divided areas that results in, when only four colours are applied, no two similarly coloured areas will touch.
Hence: the very simplistic counter-examples of a long rectangle abutting four smaller rectangles, or a cirlce interposed over four squares.
It should say:
No matter how you divide a 2D plane, you never need more than four colours to ensure that no two similarly coloured areas will touch.
: Isn't that what the first sentence already says? Dmharvey  Talk 20:47, 2 October 2005 (UTC)
replies: text: I agree, the first sentence is very ambiguous and at first glance seems to say that no more than four colors can be used to satisfy the property.
I believe it has been proven that a counterexample cannot exist...am I wrong?  Yet I see reference to that proof.
----
The problem be summarized to the following:
On a plane where you have 5 points can you connect each point to the other without intersecting any lines.
----
What is actually meant by 'plane'? I see here that points are used to represent areas and lines connecting those points are used to represent contiguity. But that is exactly what I did when as a child I figured I had 'proven' the theorem or conjecture yet my math teacher claimed that since obviously a solution a child could come up with could not be correct there must be some problem with my proof, possibly, for all he knew, it might not have been proven at that time that points connected to lines are toplogically equivalent to areas connected by contiguity. Basically he agreed that it sure did look like if points are in fact equivalent to areas and lines connecting points are actually equivalent to contiguity then sure it was indeed blatantly obvious a proof. So maybe in those days (1968 or thereabouts) it had not yet been proven that points connected by lines can validly (formally, axiomatically, theorematically, or rigorously or some such adjective) represent or correspond to areas connected by contiguities?
As a child my solution had simply been to observe that three connected points is topologically equivalent to a circle and that thus any fourth point (representing a fourth area) must be inside that circle or outside that circle or on that circle. Regardless of wether it is inside or outside, if needs three lines to connect it to the pre-existing three. It is actually irrelevant whether you picked inside or outside to place the fourth point, in either case the result is (topologically equivalent to) a tripartite circle (or a plan view of a tetrahedron). Thus in either case you have surrounded one of the four colours, freeing it up to be used again. No matter where you put the fourth point (area) you surround one of the three or place the fourth where it itself is surrounded. So always it is free to used for the next point you choose to add.
So topologically, by deforming the plane like some kind of fractal affine-transformation kind of deal (topogical sretching and deforming), it seemed pretty much obviously proven. Meanwhile if you choose to define a plane as that which is defined by three points, being as how three points on a plane defines a plane, then maybe you are looking at whether a plane need have any more than three points, and if it does where do you put them, what shape are they, and how many, exactly, are there, because you might as well colour points instead of regions unless oh wait a sec, how many points can all mutually be congruent on a plane of X number of points? Are we talking about a quantum plane, maybe using Buckminster Fuller vector equilibria as edges or points? Or a classical plane? If a classical plane does that permit topology or only Euclidean geometry?
Yeah I know, I have been way too many hours without sleep. Sorry. But still, surely how you define a plane makes a lot of difference, and when exactly did it get proven that points and lines are actually useable in proofs of plane geometry? Or is it only a proof of topology not strictly of plane geometry?
Basically for umpteen years I have yet to find anything that makes clear what is unproven or unprovable about 'the simple proof'... such as the one the originator of the conundrum had in mind maybe hmm, if they didnt have a huge computer-proof in mind; wasn't part of the whole conundrum the fact that supposedly the proof was simple? (Um, no, maybe not; I am maybe thinking of Fermat's Last as the one someone claimed was simple and elegant but didnt write in margin.)
 
What seemed simple about the points and lines method was that you show ab initio that starting from the very first area each time you add an area you automatically free up a colour each time you add an area that requires a fourth colour, so provided you correct your previous colour-scheme each time you add an area you need never get confused enough during the course of building the map to ever lose track of the fact that you never needed a fifth colour. Thus you do not find yourself looking at some hugely complicated map wondering how to colour it, you instead see step by step during construction that you have enough colours to be able to add another area if you want to.
Knotwork 11:34, 23 August 2006 (UTC)

text: I think that not every planar graph can be obtained by the method you describe. What happens to your algorithm for the graph formed by the vertices and edges of an octahedron? What colouring does it produce? Dmharvey 20:13, 23 August 2006 (UTC)

text: In short, yes, you are thinking of Fermat's Last Theorem. Most people believe that Fermat's proof was probably incorrect, like those of many others to later work on the theorem. The Four Color Theorem also had a very large number of incorrect proofs, the most notable of which stood unchallenged for over a decade. Your solution is not correct, because it's not hard to show that for certain planar graphs, you can find a greedy coloring sequence which uses an arbitrarily large number of colors. In other words, unless you constrain the greedy coloring sequence in some way, which you do not appear to do, it will not succeed in four coloring an arbitrarily planar graph. Deco 23:14, 23 August 2006 (UTC)
thread_title:  Make it clear! 

message: I believe it is true that despite the existence of several countries
that are in multiple sections, the political map of the real world
happens to be 4-colorable.  In most case the extra sections don't
add any complexity: for example, Alaska only borders Canada, which
already borders the Lower 48 States.  Ceuta only borders Morocco,
and the main part of Spain could be extended to touch Morocco without
interfering with any other borders.  And so on for most of the others;
island territories don't matter at all.
Russia and Azerbaijan do add complexity, but I think it happens to be
true that it can be worked around.  I say "I think" because I worked
it out to my satisfaction once, but I never showed it to anybody else.
Can anyone either confirm this from a reliable source, or else
disconfirm it by listing a group of adjacent countries that require
5 colors (ignoring water areas)?  I think this is a point worth
mentioning in the article.
--Anonymous, 07:20 UTC, November 16, 2005
:This is an interesting point, although I'm not sure how relevant. There's only one country contained within another as far as I know, and it doesn't pose a problem. If you include water, there are many contained bodies of water, so I'm betting you'd need 5 colours, but you're excluding those. I can't think of any counterexample, but I haven't seen a complete 4-color world map done. Deco 09:09, 16 November 2005 (UTC)
replies: text: I think this would be a great bit of trivia to have in the article.  I shall look into it.  Equalpants 00:13, 12 January 2006 (UTC)
replies: text: If you ignore the geographical features (Coloring the lakes with different colors throughout the map, Alaska blue and USA red, antarctica yellow, etc.), you are able to color the world with four colors. I'm not sure about the globe, but a 2-D map is possible. --24.149.204.116 13:48, 1 August 2006 (UTC)
replies: text: According to stereographic projection, it is possible. Yuide 10:38, 2 January 2007 (UTC)
thread_title:  Four-coloring the real world 

message: Surely there is a policy against moving articles to change between UK/US spelling? I mean, it just seems like a waste of everyone's time.... it was perfectly fine at Four color theorem, no need to move it, and I'm speaking as someone who uses british spelling habitually. Dmharvey 22:49, 24 January 2006 (UTC)
replies: text: Agreed. This contradicts policy (which is to use one spelling convention consistently in each article and not change it for no good reason) and is disruptive. Putting it back.  Deco 01:35, 25 January 2006 (UTC)

text: FYI, anonymous user User:143.210.44.211 is the culprit. I suspect they're a simple vandal up to no good. Deco 01:36, 25 January 2006 (UTC)
replies: text: I have reverted all changes by User:143.210.44.211, as well as subsequent partial fixes of those changes, and moved the page back to its original location. Deco 01:38, 25 January 2006 (UTC)
replies: text: Good stuff. (For others' reference, it was reverted to the 15th Jan version: ). Dmharvey 03:00, 25 January 2006 (UTC)
thread_title:  why the move? 

message: It seems to me that the Image:FourColorMapEx.png is not very informative. We have some nice examples in the article so I would suggestto  remove it. What do you think? kuszi 09:45, 31 March 2006 (UTC)
replies: text: I like that image and think it should stay.  It's a good introductory example of the type of objects the 4-color theorem is concerned with.  Among other things, it emphasizes the fact that the theorem is not really concerned with real-life maps. Redquark 17:44, 10 April 2006 (UTC)
replies: text:  I am convinced kuszi 21:24, 4 June 2006 (UTC)
thread_title:  Image to remove?

message: I'm pretty sure you can't color this with 4 colors.
145px|right
: Colour the outside band red, and then the five interior strips in blue, green, blue, green, blue, in that order. So actually three colours is enough. Dmharvey 01:55, 22 April 2006 (UTC)
replies: text: You got me there. Brun8 17:12, 22 April 2006 (UTC)
replies: text:  You can color that with only 3 colors.
because the person drawing the map is focused on the one large region, they fail to notice that the remaining regions can in fact be colored with two colors.
I'm sure this one doesn't work either, I just had to have a try. Any thoughts?  --Nosfartu 21:57, 15 September 2007 (UTC)

text: Of course, it works.. I wish that I could think that quickly. The trick seems to be to look for the pieces with the least neighbors, so the trick is to try to maximize the number of neighbors I think. It just makes my head hurt though. Good job! --Nosfartu 22:29, 15 September 2007 (UTC)
replies: text:  It's really not very difficult to find a solution for these superman figure, if you first ignore the regions having less than four neighbors. If you find a solution irgnoring this small regions, one color of the four is allways left to color finally the small regions too.  —Preceding unsigned comment added by 84.59.238.149 (talk) 16:48, 25 November 2007 (UTC)

text: Change the red regions bordering the purple region to green, and then change the purple region to red. Eric119 18:51, 16 September 2007 (UTC)
replies: text:  And, now create four copies of the superman figure and place it to the top, bottom, left and right of the modified chess board. --88.68.103.51 (talk) 11:47, 25 November 2007 (UTC)
thread_title:  what about this 

message: I'm thinking of adding a more detailed discussion of the specific algorithm described in "Efficiently four-coloring planar graphs" by Robertson, Sanders, Seymour, and Thomas, with enough detail to enable an implementation. Would appreciate any positive or negative feedback on this idea. Deco 19:54, 8 August 2006 (UTC)
replies: text: Update: I've described the linear-time five-coloring algorithm from the same paper at five color theorem. Deco 00:32, 9 August 2006 (UTC)
thread_title:  Describing the best known algorithm 

message: The information in these two sections is really starting to overlap.  For example, the issue of coloring water bodies is repeated with slightly different rationale in each.  What should be done to merge them? --Jonrock 23:00, 30 September 2006 (UTC)
thread_title:  "Not for Mapmakers" vs. "Real-world counterexamples" 

message: Is this article saying that with any map. you can colour it with at most 4 colours, so that the same colour doesn't touch itself? If so, is the picture of the tessellated parallelograms — which apparently needs 7 colours to colour it in — a counter-proof? I'm not exactly a mathematician, so if someone could correct me (or possibly the article?) that would be very nice.
Thanks. Ian F. 21:22, 23 October 2006 (UTC)
replies: text: That image is showing how the result changes when you go from a map on a plane to a map on a torus.  The "shape" of the space you are drawing the map on is of fundamental importance to the result, as this section on "Generalizations" demonstrates. --Jonrock 01:00, 24 October 2006 (UTC)
thread_title:  Confusing... 

message: I know many people already tried to give a proof to the theorem in a simple manner and all of that, and they have failed. Well, I´m another person who wishes to try to solve this problem, even though it´s not likely I´m gonna do any better than all of those great mathematicians who have already done it. Anyway, where can I expose my own proof, so people can say what´s wrong about it? I see Wikipedia is probably not the best place, but I can see above on the talk page many people saying they´ve colored graphics in a way they need five colors (and failed), and no one complained... Is there a website or something in which this would be more appropriate?FoCoTh 00:09, 24 December 2006 (UTC)
thread_title:  I want people to see my own proof! 

message: I recently created a page describing the Discharging method, which is one of the central ideas in the proof of the Four Color Theorem.  Is anyone willing to read it and give feedback (it's my first significant contribution to wikipedia, so I'm eager to have other people look at it).  Thanks. Ptrillian 07:30, 2 January 2007 (UTC)
thread_title:  Discharging Method 

message: could someone please disprove me that every graph of chromaticity n has a Kn minor?
It is true for n=1 to 3 (dichromatic graphs do have an edge, trichromatic graphs contain a cycle and graphs without K4 minor are either outerplanar or subdivisions of those and there's a unique (up to isomorphism) way to tricolor outerplanar triangulations and can be generalised to their subdivisions). If this was right for every n it would imply the four color theorem.
Does there exist a nonplanar graph without a K5 minor that needs 5 colors?
213.220.248.216 20:24, 9 January 2007 (UTC)
replies: text: Congratulations! You have just reconjectured Hadwiger's conjecture. As you note, it can be easily proven for n≤4. Clearly the n=5 case implies the FCT; in fact they are equivalent (or so claims the wikiarticle; I certainly can't see the proof myself). The n=6 case is also true. All other cases are open. Unfortunately, there doesn't seem to be any proof of the n=5,6 cases other than via the FCT, so this doesn't allow an easier proof. Algebraist 23:35, 18 February 2007 (UTC)

text: ps. Bollobás' Modern Graph Theory sets as an exercise that if G has chromaticity 5 then G is contractible to K5 or every vertex v of G is such that G-v is contractible to K5 with one edge missing. So presumably that result is fairly easy. Algebraist 23:38, 18 February 2007 (UTC)
thread_title:  chromaticity <= clique-minor number ? 

message: In 2000, the Indian mathematician Ashay Dharwadker discovered a new proof of the four color theorem . Since the proof is written in the traditional style and does not depend on computer calculations, I suggest that the main article should be rewritten. 122.163.105.79 14:36, 14 May 2007 (UTC)
replies: text: Thanks for bringing that to our attention. I corrected the Spanish article. -- Jitse Niesen (talk) 13:09, 23 May 2007 (UTC)
replies: text: I did, too. (See corrected article) -- Cullinane 13:53, 23 May 2007 (UTC)

text: The user "Rstewart" (very likely a sock puppet of Dharwadker) on June 23, 2007, removed a  disputed-article template from the Spanish article. For comments on June 23 by "Rstewart," see the Spanish article's discussion. --Cullinane 00:16, 25 June 2007 (UTC)
thread_title:  A new proof of the four color theorem 

message: Anonymous user 122.163.102.246 on May 31, 2007, vandalized 25 math articles in Wikipedia.  Note that the first six digits of the user's IP address agree with the New Delhi, India, IP address of some recent comments here in "A new proof of the four color theorem." For further details, see User talk:Cullinane.
-- Cullinane 01:25, 1 June 2007 (UTC)
thread_title:  Four-color-related vandalism 

message: Is there any possibility of including some intuitive description of the proof? I appreciate the original proof relied in part on computer work, but there must have been *some* intuition necessary to get to that point, right?
: There is no reliable proof, since the theorem is simply wrong. --Fsswsb 19:27, 24 October 2007 (UTC)
replies: text: Note that this has been discussed on User talk:Fsswsb#Four color theorem.  (Fsswsb has been changing the article to show a 5-color image as "proof" that the theorem is wrong, even after I've posted the 4-color solution for it to his own user talk page and he replied to it.)  You can see some examples and explanation there if you wish. --Closeapple 19:42, 24 October 2007 (UTC)
thread_title:  Description of the Proof 

message: Is it normal within Wikipedia to use US english spellings? I know Wikipedia is very US-centric, so from that point of view the answer is yes. However, standard english would require standard english spelling i.e. 'colour'. Any consensus, bearing in mind the world doesn't begin and end at the US border?  —Preceding unsigned comment added by Fizzackerly (talk • contribs) 12:07, 20 September 2007 (UTC)
replies: text: The consensus is to use whatever the first major contributor used; see Wikipedia:Manual of Style#National varieties of English. -- Jitse Niesen (talk) 12:18, 20 September 2007 (UTC)
thread_title:  'color' or 'colour' 

message: The two top diagrams are the purported disproofs added by User:Fsswsb, the two bottom diagrams are correctly colored by me. -- ArglebargleIV 22:38, 1 November 2007 (UTC)
thread_title:  Purported disproofs 

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thread_title: Fair use rationale for Image:ColorXXX.png