Color-induced subgraphs of Grünbaum colorings of triangulations of the sphere

A Grünbaum coloring of a triangulation is an assignment of colors to edges so that the edges about each face are assigned unique colors. In this paper we examine the color induced subgraphs given by a Grünbaum coloring of a triangulation and show that the existence of connected color induced subgraphs is equivalent to the Three Color Theorem. A triangulation of an orientable surface is an embedding of a simple graph in that surface such that the boundary of every face is a 3-cycle. Grünbaum [3] conjectured the following: Conjecture. For every triangulation of each orientable surface it is possible to color the edges by three colors in such a fashion that the edges of each triangle have three different colors. Following Archdeacon [1], we will call such colorings Grünbaum colorings. We will use the colors R, S, and B for Red, Silver and Black. Definition 1. A Grünbaum coloring of a triangulation G is a map γ from the set of edges of G to the set {R, S, B} such that the edges around a face of G are each given distinct colors. A Tait coloring of a 3-regular graph is a 3-coloring of the edges so that no two adjacent edges share the same color. It is not hard to see that a Grünbaum coloring of a plane triangulation corresponds to a Tait coloring of the dual graph. In fact, by using the equivalence of the Four Color Theorem with the statement that every simple 2-edge connected 3-regular planar graph has a Tait coloring, it can be shown that Grünbaum's conjecture implies the Four Color Theorem for triangulations of the sphere. Thus Grünbaum's conjecture, if true, would be a strengthening of the