A Boolean proof of Richardson's theorem on graph theory

Abstract

The demonstrations of many graph-theoretical properties can be reduced to the proofs of certain Boolean identities. A Boolean equation AX = X with an unknown Boolean n -vector X , its negation X and an assigned ( n x n )-Boolean matrix A is considered. Necessary and sufficient conditions (Theorems 1 and 2) and a sufficient condition (Theorem 3) that the equation AX = X has solutions are obtained. Then it is proved that the graph-theoretical interpretation of the proposition of this theorem 3 is equivalent to Richardson's Theorem (i.e., a graph without odd circuits has a kernel) and the proof is a Boolean proof of Richardson's theorem.