Solvability of Cubic Graphs - From Four Color Theorem to NP-Complete

Abstract: Similar to Euclidean geometry, graph theory is a science that studies figures that consist of points and lines. The core of Euclidean geometry is the parallel postulate, which provides the basis of the geometric invariant that the sum of the angles in every triangle equals $\pi$ and Cramer's rule for solving simultaneous linear equations. Since the counterpart of parallel postulate in graph theory is not known, which could be the reason that two similar problems in graph theory, namely the four color theorem (a topological invariant) and the solvability of NP-complete problems (discrete simultaneous equations), remain open to date. In this paper, based on the complex coloring of cubic graphs, we propose the reducibility postulate of the Petersen configuration to fill this gap. Comparing edge coloring with a system of linear equations, we found that the postulate of reducibility in graph theory and the parallel postulate in Euclidean geometry share some common characteristics of the plane. First, they both provide solvability conditions on two equations in the plane. Second, the two basic invariants of the plane, namely the chromatic index of bridgeless cubic plane graphs and the sum of the angles in every triangle, can be respectively deduced from them in a straightforward manner. This reducibility postulation has been verified by more than one hundred thousand instances of Peterson configurations generated by computer. Despite that, we still don't have a logical proof of this assertion. Similar to that of the parallel postulate, we tend to think that describing these natural laws by even more elementary properties of the plane is inconceivable.