The Planar Hajós Calculus for Bounded Degree Graphs

Abstract

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.