Abstract
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 χvef(G). In 1993, Borodin established the tight upper bound of χvef(G) to be Δ+2 for plane graphs with maximum degree Δ≥12. In 2011, Wang and Zhu asked: what is the smallest integer Δ0 such that every plane graph with Δ≥Δ0 is entirely (Δ+2)-colorable? For the initial step to determine the exact value of Δ0, Borodin asked in 2013: is it true that χvef≤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.
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