Secondly, bizarre maps with regions of finite area and infinite perimeter can require more than four colors.

The example of the complete graph K, which is 1-planar, shows that 1-planar graphs may sometimes require six colors.

To color the icosahedron, such that no two adjacent faces have the same color, requires at least 3 colors.

In addition, bizarre maps (using regions of finite area but infinite perimeter) can require more than four colors.

If the four-color conjecture were false, there would be at least one map with the smallest possible number of regions that requires five colors.

The Szilassi polyhedron is an example that requires seven colors.

This is the best possible result of this type, as the 5-cycle requires three colors but has exactly 2n/5 neighbors per vertex.

The Petersen graph has chromatic index 4; coloring the edges requires four colors.

A snark is a bridgeless cubic graph that requires four colors in any edge coloring.

Then G requires at least three colors in any coloring, but has no triangle, so it is not perfect.