Abstract: The arguments of the least colors that should be used to dye the 2D net color block
graph, such as maps and patterns, have puzzled the academia for one hundred and sixty years or
more. The basic reason is that the scholars have been working on to solve the problem with
classical mathematics methods, even though it is not a classical mathematics problem. Visualized
engineering geometry is used to turn the 2D net color block graph into dyeing equivalence normal
map, and critically adjacent domain axioms proved. The smallest unit of maps and their dyeing
mode is also established. Two kinds of map dyeing mode are found in the process. The first is
simple dyeing mode, and the second is the relevant dyeing mode. At the same time, a visualized
method is developed combining the dyeing and the process of turning the smallest map unit into
maps together. It is proven critically that any simple maps are suitable for the least four colors
dyeing method, but for the relevant maps, the number of colors that should be used are uncertain.

Key words: normal map, dyeing equivalence, simple map, adjacent domain, reducing rule