Vertex-Edge-Face Coloring of Planar Graphs

Abstract

In this paper the new coloring of planar graphs, vertex‐edge‐face coloring, will be introduced. A vertex‐edge‐face coloring of a simple planar graph G is a proper coloring of all elements, including vertices, edges and faces of G. We will give two conjectures for the upper bound of vertex‐edge‐face and list vertex‐edge‐face coloring of a simple planar graph. However, we will prove these conjectures for planar graphs with a maximum degree of at least 12.