The discharging technique is essential to the proof of the 4 Color
Map Theorem. A *charge* is applied to the vertices (or edges,
or triangles) in such a manner that
the total charge is known to be positive. The charge is then
redistributed in such a way that the overall charge remains the same
(and positive). Since the charge is positive there must be a
configuration with a positive charge. By examining all ways in which
positive charges can arise in the redistribution process a set of
unavoidable configurations can be constructed.

Above is the first argument in this talk making use of discharging. The result itself can of course be obtained more directly:

Suppose each vertex is of degree at least 6. Since every edge has two endpoints, the total number of edges is at least

3V = 3V_{B}+ 3V_{I}

On the other hand, according to the combinatorics of a triangulation the total number of edges is

which is a contradiction.E = E

_{I}+ E_{B}= V_{B}+ 3V_{I}- 3 + V_{B}< 3V_{B}+ 3V_{I}

[21-Apr-1997]