Clifford algebra and 4-color theorem (Q1921268)

The 4-color theorem was shown to be equivalent to a statement of bracketing of cross products of orthogonal vectors in three space by Kaufmann. This paper establishes a constructive proof of the 4-color theorem by employing Clifford algebras. The associative Clifford product can be used to model the cross product which renders bracketing to be obsolete. The proof proceeds as follows: (1) If two bracketings of cross products of \(n\) orthogonal vectors are nonzero, then they are identical. (2) The nature and number of inequalities obtained from bracketing are studied, which results in \(2n-3\) inequalities to be satisfied. (3) The existence of a nonzero choice of such orthogonal vectors is shown, which completes the proof.