How was the 4 colour theorem proved?
. A computer-assisted proof of the four color theorem was proposed by Kenneth Appel and Wolfgang Haken in 1976. Their proof reduced the infinitude of possible maps to 1,936 reducible configurations (later reduced to 1,476) which had to be checked one by one by computer and took over a thousand hours .
Is the four color theorem solved?
The four-colour problem was solved in 1977 by a group of mathematicians at the University of Illinois, directed by Kenneth Appel and Wolfgang Haken, after four years of unprecedented synthesis of computer search and theoretical reasoning.
Why is the four colour theorem important?
The 4-color theorem is fairly famous in mathematics for a couple of reasons. First, it is easy to understand: any reasonable map on a plane or a sphere (in other words, any map of our world) can be colored in with four distinct colors, so that no two neighboring countries share a color.
Who discovered the four color theorem?
Though there were other proposed proofs of the time, namely those written by Baltzer (1885) and Peter Guthrie Tait (1880), Kempe was given credit as the one who proved the four-color theorem.
Who finally proved the four color theorem?
The four color theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken after many false proofs and counterexamples (unlike the five color theorem, proved in the 1800s, which states that five colors are enough to color a map).
What is the four color theorem?
The four-color theorem states that any map in a plane can be colored using four-colors in such a way that regions sharing a common boundary (other than a single point) do not share the same color. This problem is sometimes also called Guthrie’s problem after F. Guthrie, who first conjectured the theorem in 1852.
Is every four colorable graph planar?
The Four Color Theorem states that every planar graph is properly 4-colorable. Moreover, it is well known that there are planar graphs that are non-4 -list colorable. In this paper we investigate a problem combining proper colorings and list colorings.
Which colour is used in the map for the plane region?
Explanation: Various colours are used for the same purpose. For example, generally blue is used for showing water bodies, brown for mountain, yellow for plateau and green is used for plains.
What is the proof of the four color theorem?
Early proof attempts. Tait, in 1880, showed that the four color theorem is equivalent to the statement that a certain type of graph (called a snark in modern terminology) must be non- planar. In 1943, Hugo Hadwiger formulated the Hadwiger conjecture, a far-reaching generalization of the four-color problem that still remains unsolved.
Is Kempe’s 5-color theorem still valid?
The proof appeared valid for 11 years until Percy John Heawood published a paper pointing out that in a particular subcase, Kempe’s method will not yield a 4-coloring (see here for more detail). Kempe tried to revise his proof, but was unable to do so. Note: The proof of the 5-color theorem using Kempe chains that we showed above is still valid.
How many colors do you need to prove 6-colorability?
Since the 4-color theorem is rather difficult to prove, let us start with the substantially easier (and weaker) 6-color theorem: no map requires more than 6 colors to ensure that no two adjacent regions have the same color. We can apply theorems about planar graphs in order to prove the 6-colorability of all maps.
Do we really need four colors in a neighborhood?
We never need four colors in a neighborhood unless there be four counties, each of which has boundary lines in common with each of the other three. Such a thing cannot happen with four areas unless one or more of them be inclosed by the rest; and the color used for the inclosed county is thus set free to go on with.