Plane Graphs with Maximum Degree 8 Are Entirely (+3)-Colorable

Let G=(V,E,F) be a plane graph with the sets of vertices, edges, and faces V, E, and F, respectively. If one can color all elements in VEF using k colors so that any two adjacent or incident elements receive distinct colors, then G is said to be entirely k-colorable. Kronk and Mitchem [Discrete Math 5 (1973) 253-260] conjectured that every plane graph with maximum degree is entirely (+4)-colorable. This conjecture has now been settled in Wang and Zhu (J Combin Theory Ser B 101 (2011) 490501), where the authors asked: is every simple plane graph GK4 entirely (+3)-colorable? In this article, we prove that every simple plane graph with 8 is entirely (+3)-colorable, and conjecture that every simple plane graph, except the tetrahedron, is entirely (+3)-colorable. (C) 2012 Wiley Periodicals, Inc.