Plane Graphs with Maximum Degree Δ ≥ 8 Are Entirely ( Δ + 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 V ∪ E ∪ F 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 Δ is entirely ( Δ + 4 ) ‐colorable. This conjecture has now been settled in Wang and Zhu (J Combin Theory Ser B 101 (2011) 490–501), where the authors asked: is every simple plane graph G ≠ K 4entirely ( Δ + 3 ) ‐colorable? In this article, we prove that every simple plane graph with Δ ≥ 8 is entirely ( Δ + 3 ) ‐colorable, and conjecture that every simple plane graph, except the tetrahedron, is entirely ( Δ + 3 ) ‐colorable.