Theorem
Using Kempe’s method of color exchanges in so-called Kempe Chains, any counter-example to his proof of the Four Color Conjecture can be properly colored with at most four colors.
Proof
The following proof requires reference to an actual map generally accepted as a counter-example disproving Kempe’s proof. Because of their historical significance, and because of their differences in complexity, the maps by Errera (Fig. 1.0) and Heawood (Fig. 2.0) have been selected as exemplars that exhibit all the properties required of any map offered as a counter-example to Kempe’s proof.
(click on picture for larger view)
Using Kempe’s method of color exchanges in so-called Kempe Chains, any counter-example to his proof of the Four Color Conjecture can be properly colored with at most four colors.
Proof
The following proof requires reference to an actual map generally accepted as a counter-example disproving Kempe’s proof. Because of their historical significance, and because of their differences in complexity, the maps by Errera (Fig. 1.0) and Heawood (Fig. 2.0) have been selected as exemplars that exhibit all the properties required of any map offered as a counter-example to Kempe’s proof.
(click on picture for larger view)
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