Polychromatic colorings of bounded degree plane graphs

A polychromatic k-coloring of a plane graph G is an assignment of k colors to the vertices of G such that every face of G has all k colors on its boundary. For a given plane graph G, one seeks the maximum number k such that G admits a polychromatic k-coloring. In this paper, it is proven that every connected plane graph of order at least three, and maximum degree three, other than K ₄ or a subdivision of K₄ on five vertices, admits a 3-coloring in the regular sense (i.e., no monochromatic edges) that is also a polychromatic 3-coloring. Our proof is constructive and implies a polynomial-time algorithm.