1. In progress : New Approach to construct $MPG5$.
2. Tools to compare, test and search MPG5 particularities by starting with PLANTRI.
3. About 4CT: Another formulation of the Four Color Theorem
4. Conferences : ISICT 06/2003 | SIAM Discrete Mathematics 08/2002

We are looking for a new approach in order to construct all $MPG5$. Here we will give some informations about this goal.

• Matrix (php file) of $MPG5_14$ with plan orientation.
• Transformations on $MPG5$ :
  • HowTo: pdf(LaTeX document with bitmap font!!), All Ressources
  • Demo/Read Me : V2.0.1 mpg5.html
  • Sources : mpg5_constructor-2.0.1.zip 21/06/2002

We have running PLANTRI (The authors are Gunnar Brinkmann (University of Bielefeld) and Brendan McKay (Australian National University) : This soft can be found at http://cs.anu.edu.au/people/bdm/plantri/ with the following command line "plantri40 -c3 -m5 20 -a file.txt", see command line for 17 to 26 vertices. All no-isomorphic graphs have been stored in MySql database in order to compute without restriction, all tests.

• Display MPG5 graph (between 17 and 21) : 17 | 18 | 19 | 20 | 21
  Scripts:
  • How to load a ascii PLANTRI file graphs in MySql : loading.html | Database structure | Documentations
  • How to compute, compare graphs in MySql database : studies.html | Documentations
  • How to select and display graphs particularities : select.html | Documentations
• Results : | Which graphs cannot use $T^{-1}$ and $D$ : results
• Download : scripts PHP, HTML Form, MySql structure + data n = [17,20] in ZIP format : 1500ko plantri_mpg5_php_mysql.ZIP (01/07/2002)
• Docs...

sometimes "It is necessary to complicate a problem to simplify
the solution of it" (Erdös) 

The theorem of the four colors is a
particular case of a more general theorem concerning the graphs
in space than one can formulate as follows:  That is to say N
points coloured in space.  Some of these points are dependent. 
Four points with more can be dependent two to two.  Two dependent
points are different colors.  Two nondependent points are of the
same color.  Two points of different colors are always dependent.
 Under these conditions four colors are enough to colour N points
of this graph.  Some mathematicians think that this general
theorem can be shown without computer, like his relationship to
the traditional theorem of the four colors.  See Bulletin
Association A.A.M.T.  N° 82 Août 2001. Email: AssocAAMT@aol.com

" Il faut parfois compliquer un problème pour en simplifier la
solution " ( Erdös )

Le théorème des quatre couleurs est un cas particulier d'un
théorème plus général concernant les graphes dans l'espace qu'on
peut formuler ainsi :

Soit N points colorés dans l'espace. Certains de ces points sont
liés. Quatre points au plus peuvent être liés deux à deux. Deux
points liés sont de couleurs différentes. Deux points non liés
sont de même couleur. Deux points de couleurs différentes sont
toujours liés.  Dans ces conditions quatre couleurs suffisent
pour colorer les N points de ce graphe.

Quelques mathématiciens pensent que ce théorème général peut être
démontré sans ordinateur, ainsi que sa relation avec le théorème
classique des quatre couleurs.

Voir Bulletin Association A.A.M.T. N° 82 Août 2001. Email: AssocAAMT@aol.com

Accepted paper for ISICT 06/2003 - Dublin, IR : http://www.isict.org

Accepted paper for SIAM Discrete Mathematics 08/2002 - San Diego CA : Details | http://www.siam.org/meetings/archives/index.htm#DM