UNCOLORABLE TRIVALENT GRAPHS

Abstract

The text that follows is an extended write-up of the Ross Ashby Memorial Lecture of the International Federation for Systems Research, delivered by invitation to a plenary session of the Thirteenth European Meeting on Cybernetics and Systems Research at the University of Vienna on April 10, 1996. For ease of reading I have retained the informal style of the lecture theater, but without sacrificing accuracy and rigor in my theorems and proofs. Isaacs 1975 detailed a way to make uncolorable 3-graphs. The first part of this communication shows that it is, in principle, the only way and that all such graphs that can define maps must be nonplanar, thus proving the four-color map theorem by exclusion. The second part shows that the method is ineffectual for closed chains of units with a linkage of less or more than 3, so proving that Isaacs found all uncolorable 3-graphs. The associated proof of the map theorem in no way depends on this stronger result but on the fact that chains with a triple or greater linkage, whether colorable or not, cannot be planar. The third part simplifies the Isaacs dot product, proves that his Q class has no more members, and relates the 4-color proof in the first part to the proofs in Appendix 5 of Spencer-Brown 1997.