A Derivative of the Four Color Theorem? On Coloring Lines

There is a mathematical theorem called the "Four Color Theorem."
This theorem states that "when coloring regions on a flat map divided by curves, a minimum of four colors is sufficient to ensure that no two adjacent regions share the same color." It might be somewhat well-known, as it appeared in the movie "Suspect X."

In simple terms, the Four Color Theorem is a problem asking how many colors are needed to ensure that adjacent regions do not share the same color when coloring countries or municipalities on a map.

Then, what about the case of curves rather than regions?

In short, the essence of the problem is as follows.

There are many curves on a flat map, intersecting everywhere. How many colors are sufficient to ensure that intersecting curves do not share the same color?

This question occurred to me while creating a national highway map for my imaginary map world. To make the intersections and overlaps of national highways easy to understand, I created a route map where intersecting highways are displayed in different colors. To achieve this goal, highways that do not intersect (or even graze each other) can be the same color. Since having too many colors makes it difficult to distinguish between "similar but different" colors, making the map hard to read, I preferred to represent it with as few colors as possible.

As a result, the national highway map of the imaginary map world could be colored using 8 colors.

So, can any route map be colored with 8 colors?
The answer is no.

If the center of the kanji character for "rice" (米) is the starting point and the 8 directions are routes 1 through 8, then 8 colors are needed to color them. If the center is the starting point and the 16 directions are routes 1 through 16, like the character "✺️," then 16 colors are needed to color them. As shown, since any number of lines can be drawn from a single point, there is no fixed number of colors that are "at least required to color them." In an extreme case, it is possible that as many colors as there are routes could be needed.

However, it is important to note that "the number of colors required for coloring = the maximum number of routes meeting at one point" is not necessarily true. Let's draw a circle surrounding the center of the "rice" character. If there is such a circular route, the number of colors required for coloring becomes 9, not 8.

In the Imaginary World, the maximum number of national highways meeting at one point is 7 (there are two such points, the Hokou Intersection and the Nemiya Bridge Intersection). However, even then, it cannot be colored with 7 colors. The reason is obvious if you look at the route map around Kusakabe City, where the Hokou Intersection is located.

The Hokou Intersection in the center of Kusakabe City is where a total of 7 routes meet: Route 1, which passes through the intersection, and Routes 28, 169, 172, 304, 415, and 419, which start or end at the intersection. However, there is Route 305, a circular route that surrounds this point from the outside. Since Route 305 intersects all 7 routes extending radially from Hokou, an 8th color is required. Note that Route 243, which intersects Route 305 but does not reach the Hokou Intersection, does not intersect Route 28, so it can use the same green color as Route 28.