how many colour is required to colour an arbitrary map?

Answer 1

User Rich Z is completely uninformed. While the answer is "at least four colors," and it can often be demonstrated to be four (like in his example using the USA), this statement:

"Ever since people started to deal with this issue the answer was quickly proven to be four colors."

is totally incorrect and was obviously fabricated on the spot. The theorem that proves this fact is known as the four color theorem.

This theorem was NOT proven "ever since" anything. People started making maps hundreds of years ago but never really sought to minimize the number of colors used. Not only that, but when mathematicians sought to prove it, beginning in the middle of the 1800s, they did not prove it quickly or easily.

It was proven in 1976, well over a century after it was conjectured, and involved massive amounts of computer-based proof, examining nearly 2000 different special cases.

For an extended explanation and more details about the history of this theorem, see:
http://en.wikipedia.org/wiki/Four_color_theorem

As a final note, please keep in mind that this theorem only governs maps on a plane. A sphere can be shown to be equivalent to a plane, however, if the world were donut shaped we'd require 7 colors to color a globe. Also note that this theorem is not valid for arbitrary maps if we allow non-connected countries (e.g. the US and Alaska).

The other answerers also have gross inaccuracies in their answers:

The proof was in no way controversial - the computer algorithm was not dubious in any way, and was PROVEN to have worked. No reasonable mathematician would question an algorithm that was proven to work. Many mathematicians PREFER to avoid using computers, but that is not the same as disbelieving computer-based proof.

It was not first stated in 1870, but in 1852.

Appel and Haken had over 1900 cases. The cases were not reduced to 1500 until after Appel and Haken had proven the theorem.

The Map Color Theorem is different, and is a generalization of the Four Color Theorem.


And to veeera, please note that while your sarcastic answer is not unapprciated, I believe what I said was AT LEAST FOUR COLORS. Please read more carefully before gracing us with your wit. Also, I explained that plane and spherical maps are equivalent.

Answer 2

The first answerer said correctly that the answer is four. But it is not true that it was proved quickly. The problem was probably known from a long time ago, but was first formally stated in the 1870s by Guthrie. None of the mathematicians of his day could prove it (although it's easy to convince yourself that it's true), and many false "proofs" of it were submitted over the next century.

In 1974, Appel and Haken broke the problem down into about 1500 cases, then used a computer to methodically check each case, and proved the Four Colour Theorem that way. This is an interesting development in the philosophy of mathematics, because it was the first time that a computer was used in an essential way to prove a theorem. It's interesting because due to the massive number of cases to be checked, it would be impossible for a human to do what the computer does. So accepting the proof comes down to believing that the computer worked properly and the programme had no errors.

Although some amount of controversy does remain, most mathematicians today accept the computer proof of the Four Colour Theorem.

Answer 3

Map of a country showing states or map of countries is not a geometrical figure - figure with some specific shapes. The countries or states of a country always have irregular - non-geometrical shapes. For the identification of each matter presented on a map must have different colour from its neighbouring matter - in fact it should be presented in significantly different colour. If the map is showing agriculture zones or land and mountains - it may have minimum two colours, and most things can be presented with three colours. If you mean a map of countries where all countries are contiguous and you want to colour them so that two adjacent countries do not have the same colour, then you will need minimum of four colours. The fact that five colours are enough to colour any map is proved by studying large number of maps and most texts on graph theory. It is much easy to prove logically that five colours are enough than proving that only four are enough for map.
There are several states in the map of United States of America and in the map of India, where one state is surrounded by large number of states - six or seven. These maps can also be presented with four colours

Answer 4

If you mean a map in the plane (that is, it can be drawn on paper) where all countries are contiguous (that is, they have only one piece), and you want to color them so that no two adjacent countries have the same color, then the answer is four. This is known as the "Four-Color Map Theorem," or simply the "Four-Color Theorem."

The Four-Color Theorem was proved in the 1970s. Its proof was controversial because it involved using a computer to go through a large number of possible cases--cases which had not all been checked by a human. Some people questioned whether the proof was valid if it had to appeal to a computer program as its evidence.

The fact that five colors are enough to color any map is proved in most introductory texts on graph theory. (To prove that five colors are enough is much easier than proving that only four are enough.)

Answer 5

Four. Ever since people started to deal with this issue the answer was quickly proven to be four colors. For a good example look at a US map where the state of Colorado touches three other states in one corner and see the four colors used there.


z

Answer 6

Though cheeser nicely explained the theory, none of them read the question completely.
The question does not contain the word "minimum number of " and hence the question itself is not clear.
secondly the word "arbitrary" refers to any map plane or spherical.
in that case, unless the question is specific, no one could give best answer.
hence none of the answers given here could be rated as best.