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, we seek the maximum number k such that G admits a polychromatic k-coloring. We call a k-coloring in the classical sense (i.e., no monochromatic edges) that is also a polychromatic k-coloring a strong polychromatic k-coloring. In this paper, it is proven that every plane graph of maximum degree three, other than K4 (the complete graph on four vertices), admits a strong polychromatic 3-coloring. This initiates the study of strong polychromatic colorings of plane graphs. Moreover, our proof is constructive and implies a polynomial-time algorithm.