The Complexity of the Haj6s Calculus

The Haj6s Construction is a simple, nondeterministic procedure for generating the class of graphs that are not 3-colorable. Mansfield and Welsh have posed the problem of proving whether or not there exists a polynomial-size Haj6s construction for every non3-colorable graph. The main result of this paper is a proof that the Haj6s calculus is polynomially-bounded if and only if extended Frege proof systems are polynomially bounded. This result links an open problem in graph theory to an important open problem in the complexity of propositional proof systems. In addition, we establish an exponential lower bound for a strong subsystem of the Haj6s calculus. Lastly, we discuss an interesting graph-theoretical consequence of this result.