The four-colour mapping theorem states that all maps can be four-coloured (adjacent regions receive distinct colours, and four different colours are used in total). However, the technical definition of "map" implies that regions must be contiguous, and, in fact, it's easy to construct counter-examples to a proposed "four-colour not-necessarily-contiguous-mapping conjecture". For example:
where the two green areas constitute a single non-contiguous region. This is clearly not four-colourable since each of the five internal regions is adjacent to every other (forming a $K_5$ in the graph-theoretic sense).
Some real-world maps have disjoint components (e.g. USA, including Alaska, Hawaii), which leads me to:
Question: What is an example of a real-world map (which necessarily must have at least one non-contiguous region) that is not four-colourable?