The four-color theorem is often expressed in different ways which are loosely claimed to be equivalent. However, not all of these formulations are actually equivalent, and not all of them are even true. This means that there are "counterexamples" to certain (incorrect) formulations of the four-color theorem.

The mathematically true statement of the theorem is:

(1) Any loopless planar graph can be colored with (at most) four colors such that no edge connects two vertices of the same color.

However, the theorem is usually introduced in the context of coloring maps, and is loosely framed as saying that "Any map can colored such that no two adjacent countries have the same color." However, making this formulation mathematically precise is surprisingly challenging. For example, here is a seemingly reasonable attempt to formalize the "map" version of the four-color theorem:

(2) Let $D$ be an open subset of the plane, and consider an arbitrary partition of $D$ into path-connected open subsets $S_i$ and their shared boundaries. It is possible to color each subset $S_i$ with one of four colors, such that the shared boundary of any two subsets $S_i$ that are assigned the same color consists only of isolated points.

However, it turns out that not only is proposition (2) not equivalent to proposition (1), but in fact (1) is true and (2) is false. Indeed, even even we further require that the boundaries of the subsets $S_i$ described in (2) consist only of straight line segments and right angles, the claim is still false. A counterexample - a partition of a rectangle into six subsets satisfying the requirements of (2) that cannot be four-colored - is given in https://www.jstor.org/stable/3647828.

This "counterexample" to the four-color theorem - really a counterexample to the incorrect version (2) - demonstrates the utility of formulating the theorem in terms of graph theory, where its statement is quite simple, rather than in terms of the motivating "map" version (which can be done, but requires a large number of fairly complex conditions).