Four-color theorem

Amusingly, the proof" utilizes the famous" " four-color theorem —indeed the proof follows readily from comments" over a century ago. The present result thence indicates resonance is always conceivable for such carbon cages. Thus, a graphic resonance-theoretic approach seems a possibility such as indeed we have pur-... 

An incidental factoid. You probably know about the four-color theorem on a plane or spherical surface, four colors suffice to draw a map in such a way that regions sharing a common boundary have different colors. On the surface of a torus it takes seven colors. 

In 1943, Hadwiger stated a conjecture closely related to the four-color theorem. Recall that a graph H is called a minor of another graph G if H can be obtained from a subgraph of G by a sequence of edge-contractions. 

Th98] R. Thomas, An update on the Four-Color Theorem, Notices Amer. Math. 

A special place in the theory of vertex-colorings of a graph is occupied by the so-called four-color problem the question whether there is a four-coloring of a planar map such that every pair of coimtries that share a (nonpoint) boundary segment receive different colors. Let us show the weaker five-color theorem. Before we can prove it, we need a standard fact, which is a special case of the Euler-Poincare formula. 

The apparently first proof, offered by A. Kempe in 1880, [Kem79], turned out to be false, as did many later ones. The flaw was noticed in 1890 by P. Heawood, [Hea90], who also proved the weaker five-color theorem. Important contributions to the four-color problem, were made (among others) by G. Birkhoff and H. Heesch, [Hee69]. The original announcement of the resolution of the four-color problem by Appel Haken can be found in [AH76], and [AH89] is the last reprint. The reference to the work of Robertson, Sanders, Seymour Thomas on this subject is [RSST],... 

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