Appel and Haken published in 1977 from the University of Illinois at Urbana- Champaign a computer-assisted proof that four colors are sufficient. However, some mathematicians do not accept it because it utilized examination of cases assisted by a computer, with the possibility of software errors always remaining. On the other hand, no flaws have yet been found, in spite of repeated attempts. The first independent proof of the four color theorem was constructed by Robertson et al. in 1996 and by Thomas in 1998. Then, in December 2004, G. Gonthier in Cambridge, England -colaborating with B. Werner of INRIA in France - announced that they were able to validate the Robertson et al. (1996) proof of the color theorem by formulating it in the equational logic program called “Coq” (gaelic ?), and then were able to confirm the validity of each of its steps as reported by Devlin in 2005 and Knight in 2005.
The Heawood conjecture is a more general proposition for map coloring, stating that in a genus 0 space, including both the sphere or plane, four colors would suffice.
On the other hand, for any genus
, Ringel and Youngs were able to prove in a report published in 1968 the following theorem:
where the right-hand-side is called the floor function.
A closely related theme is that of graph coloring using computer algorithms.
The chromatic number of a graph
or
is the smallest, or minimum number of colors needed to color the vertices of a graph
so that no two adjacent vertices share the same color (p. 210 in ref. [7]); this is the smallest value of possible to obtain a
-coloring.
Let
be the graph whose vertices are the points of the plane and two points are being joined by a graph edge if they are at a distance
one from each other. Then the coloring question is: “What is the chromatic number of this graph?” This was one of Paul Erdös's favorite problems. His results are:
Then, the chromatic numbers of
are
.
More generally, the chromatic number of a graph
can be computed as the smallest positive integer
such that the chromatic polynomial
takes only positive values; thus, calculating the chromatic number of a graph is an NP-complete problem (as shown by Skiena in ref. SkS90,on pp. 211-212), but no general algorithm has been found yet for any arbitrary graph as suggested by Harary in 1994 (on p. 127 in ref.[5]). Erdös proved in 1959 (ref. [3]) that there are graphs with arbitrarily large girth and chromatic numbers (cited in ref. [1]).
Chromatic numbers and minimal colorings for many colored graphs are readily illustrated by employing Mathematica
as shown at the mathworld website.
As an example, the chromatic number can be digitally computed using ChromaticNumber
in the Mathematica
package “Combinatorica”; minimal coloring can also be computed by using MinimalColoring
in the same package. Pre-computed chromatic numbers are readily available for most remarkable or special-property graphs can be obtained using
,
(with
).
Sloane, N. J. A. Sequences
in The On-Line Encyclopedia of Integer Sequences.
Weisstein, Eric W. ``Chromatic Number.'', In MathWorld-A Wolfram Web Resource.
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