An inductive hypothesis is offered concerning the 4-color reducibility within a planar graph where the boundary between a proper subgraph and its complement has 4 points. Where such a graph having a boundary with 4 points is fully triangulated, the boundary points will form a square: the two components can be completed by connecting opposite corners of the square, and the opposite corners must be assigned different colors. The inductive hypothesis shows that (1) graphs smaller than the original graph can be 4-colored, (2) each of the components will have several possible colorings, and (3) there will always be a pair of colorings for the components that will match.
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