Structural theorem on plane graphs with application to the entire coloring number
Journal of Graph Theory
Graph Theory
Entire colouring of plane graphs
Journal of Combinatorial Theory Series B
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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 @g"v"e"f(G). In 1993, Borodin established the tight upper bound of @g"v"e"f(G) to be @D+2 for plane graphs with maximum degree @D=12. In 2011, Wang and Zhu asked: what is the smallest integer @D"0 such that every plane graph with @D=@D"0 is entirely (@D+2)-colorable? For the initial step to determine the exact value of @D"0, Borodin asked in 2013: is it true that @g"v"e"f@?13 holds for every plane graph with @D=11? In this paper, we prove that every plane graph with maximum degree @D=10 is entirely (@D+2)-colorable.