  A planar graph can be embedded in a piecewise linear manifold, and the
lattice on each linear piece can be colored with 3-coloring. If a planar graph
can be colored with multiple 3-coloring, i.e. coloring the graph in pieces with
different 3-color subsets of 4 colors, then the graph is 4-colorable. In this
paper, multiple 3-coloring was introduced, and then the combination and
partition of planar graphs for multiple 3-coloring was studied. The study
reveals that planar graphs can generally be decomposed into independent
subgraphs, and each subgraph can be triangulated into a symmetric structure for
multiple 3-coloring.
