Building a Maximal Independent Set for the Vertex-coloring Problem on Planar Graphs

We analyze the vertex-coloring problem restricted to planar graphs and propose to consider classic wheels and polyhedral wheels as basic patterns for the planar graphs. We analyze the colorability of the composition among wheels and introduce a novel algorithm based on three rules for the vertex-coloring problem. These rules are: 1) Selecting vertices in the frontier. 2) Processing subsumed wheels. 3) Processing centers of the remaining wheels. Our method forms a maximal independent set S₁ ⊂ V (G) consisting of wheel's centers, and a maximum number of vertices in the frontier of the planar graph. Thus, we show that if the resulting graph G′ = (G − S₁) is 3-colorable, then this implies the existence of a valid 4-coloring for G.