Four-Color Problem - Part II

Alfred Kempe, who showed that a minimum uncolorable map could not have a face with four edges [see Part I], gave a flawed proof that it couldn't contain a face with five edges which would then prove the conjecture. Eleven years later the flaw was discoved. His proof was an extension of his proof for four edges. Since there is one more face than colors one color must be used twice. (All four must be used at least once, of course.) In the diagram I have chosen color b. The a-d region containing A must contain D or a and d could be switched in that region leaving a for the center. Similarly for the a-c region containing A and C. Then the b-d region containing B cannot contain D and the b-c region containing E cannot contain C. The switching of b and d in the first and b and c in the second makes color b available for the center. The problem is that the first switch changes the condition required for the second switch.

In Part III I will describe my unsuccessful attempts to get around this problem.