Edge‐face chromatic number and edge chromatic number of simple plane graphs

Given a simple plane graph G , an edge‐face k ‐coloring of G is a function ϕ : E ( G ) ∪ F (G) →  {1,…, k } such that, for any two adjacent or incident elements a , b ∈ E ( G ) ∪ F ( G ), ϕ( a ) ≠ ϕ( b ). Let χ e ( G ), χ ef ( G ), and Δ( G ) denote the edge chromatic number, the edge‐face chromatic number, and the maximum degree of G , respectively. In this paper, we prove that χ ef ( G ) = χ e ( G ) = Δ( G ) for any 2‐connected simple plane graph G with Δ ( G ) ≥ 24. © 2005 Wiley Periodicals, Inc. J Graph Theory