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A planar graph can be embedded in a piecewise linear manifold, and thelattice on each linear piece can be colored with 3-coloring. If a planar graphcan be colored with multiple 3-coloring, i.e. coloring the graph in pieces withdifferent 3-color subsets of 4 colors, then the graph is 4-colorable. In thispaper, multiple 3-coloring was introduced, and then the combination andpartition of planar graphs for multiple 3-coloring was studied. The studyreveals that planar graphs can generally be decomposed into independentsubgraphs, and each subgraph can be triangulated into a symmetric structure formultiple 3-coloring.

Multiple 3-Coloring, an Approach to 4-Coloring of Planar Graphs