bill wrote: > >Okay. > >Finite planar graphs are 4-C because they cannot >contain subgraphs isomorphic to K5! > >My thinking is thus. In any K4 subgraph one vertex is >unavailable. Therefore, there is no way to force any set of >4 vertices to have four different colors. Tangled chains are >not due to the congiguration of the graph but result from a >fortuitous coloring. Plus, if two chains are tangled; there >will be two that are not tangled. > >If there are no forceable sets of four vertices, the graph >must be 4 colorable.
Except that I've already shown you a counterexample.