Four-Color Reducibility of Planar Graphs Containing Subgraphs with Four-Point Boundaries.
It is known that a planar graph containing a proper subgraph with only three points in the boundary between the subgraph and its complement is four-color reducible; that is, if the subgraph and the complement plus the boundary are each colorable with four colors, then the original graph is four-colorable. A proof of reducibility has been lacking for the case of subgraphs whose boundaries contain four points. This paper demonstrates, under an inductive hypothesis that graphs smaller than the original graph can be four-colored, that each of the components will have several possible colorings, and there will always be a pair of colorings for the components that will match. 4 pp.