The Planar Hajós Calculus for Bounded Degree Graphs

The planar Hajós calculus (PHC) is the Hajós calculus with the restriction that all the graphs that appear in the construction (including a final graph) must be planar. The degree-d planar Hajós calculus (PHC(d)) 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(d+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.