Entire -colorability of plane graphs

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 with k colors so that any two adjacent or incident elements receive distinct colors, then G is said to be entirely k -colorable. The smallest integer k such that G is entirely k -colorable is denoted by χ v e f ( G ) . In 1993, Borodin established the tight upper bound of χ v e f ( G ) to be Δ + 2 for plane graphs with maximum degree Δ ≥ 12 . In 2011, Wang and Zhu asked: what is the smallest integer Δ 0 such that every plane graph with Δ ≥ Δ 0 is entirely ( Δ + 2 ) -colorable? For the initial step to determine the exact value of Δ 0 , Borodin asked in 2013: is it true that χ v e f ≤ 13 holds for every plane graph with Δ = 11 ? In this paper, we prove that every plane graph with maximum degree Δ ≥ 10 is entirely ( Δ + 2 ) -colorable.