  For the four-color theorem that has been developed over one and half
centuries, all people believe it right but without complete proof convincing
all1-3. Former proofs are to find the basic four-colorable patterns on a planar
graph to reduce a map coloring4-6, but the unavoidable set is almost limitless
and required recoloring hardly implements by hand7-14. Another idea belongs to
formal proof limited to logical operation15. However, recoloring or formal
proof way may block people from discovering the inherent essence of a coloring
graph. Defining creation and annihilation operations, we show that four colors
are sufficient to color a map and how to color it. We find what trapped
vertices and boundary-vertices are, and how they decide how many colors to be
required in coloring arbitrary maps. We reveal that there is the fourth color
for new adding vertex differing from any three coloring vertices in creation
operation. To implement a coloring map, we also demonstrate how to color an
arbitrary map by iteratively using creation and annihilation operations. We
hope our hand proof is beneficial to understand the mechanisms of the
four-color theorem.
