The planar Hajos calculus (PHC) is the Hajos calculus with the restriction that all the graphs that appear in the construction (including a final graph) must be planar. The degree-d planar Hajos calculus (PHC(dd)) is PHC with the restriction that all the graphs that appear in the construction (including a final graph) must have maximum degree at most d. We prove the followings: (1) If PHC is polynomially bounded, then for any d ≥ 4, PHC(dd+2) can generate any non-3-colorable planar graphs of maximum degree at most d in polynomial steps. (2) If PHC can generate any non-3-colorable planar graphs of maximum degree 4 in polynomial steps, then PHC is polynomially bounded.

The Hajos calculus is a nondeterministic procedure which generates the class of non-3-colorable graphs. If all non-3-colorable graphs can be constructed in polynomial steps by the calculus, then NP = co-NP holds. Up to date, however, it remains open whether there exists a family of graphs that cannot be generated in polynomial steps. To attack this problem, we propose two graph calculi PHC and PHC* that generate non-3-colorable planar graphs, where intermediate graphs in the calculi are also restricted to be planar. Then we prove that PHC and PHC* are sound and complete. We also show that PHC* can polynomially simulate PHC.