Mathematical_Recreations-Kraitchik-2e

Geometric Recreations ~11
closed -- that is, it can be traversed completely just once
without jumping, and the path stops where it started. In
this case the path may be started at any point of the graph.
If the graph has two odd vertices it is still unicursal, but the
path starts at one odd vertex and ends at the other.
The square with its diagonals is not unicursal, but the
heptagon with its transversals is unicursal and closed. The
square with one diagonal is unicursal, but the path must
start at one extremity of the diagonal and end at the other.
2. Probably the most famous unsolved problem after
Fermat's last theorem, and certainly the most deceptively
simple in appearance, is the map-coloring problem. Suppose
that we wish to print a map of certain countries, and we want
to be sure that no two countries which have a common boundary
line have the same color. We suppose that every country
is in one piece and has no holes in it. No harm will be
done if two countries which have no common boundary have
the same color. What is the least number of colors which will
suffice for every such map?
Two colors suffice to color the squares of a chessboard,
and three for the hexagons in Figure 75. But if one wishes
to color the "ocean" around the hexagons, four colors are
needed. Thus a printer of maps must have at least four colors
at his disposal.
Heawood has shown that five colors are sufficient for every
map on the plane or on the sphere, but no one has ever produced
a map which needs that many colors, nor has it ever
been shown that four colors will always suffice.
One of the most tantalizing aspects of the problem is the
fact that the problem has been solved for surfaces more complicated
than the surface of the sphere. For example, seven
colors are necessary and sufficient to color every map drawn
on the surface of a torus. Only in the apparently simplest
case has the problem remained unsolved.