The edge-face coloring of graphs embedded in a surface of characteristic zero

Let G be a graph embedded in a surface of characteristic zero with maximum degree Δ . The edge-face chromatic number χ e f ( G ) of G is the least number of colors such that any two adjacent edges, adjacent faces, incident edge and face have different colors. In this paper, we prove that χ e f ( G ) ≤ Δ + 1 if Δ ≥ 13 , χ e f ( G ) ≤ Δ + 2 if Δ ≥ 12 , χ e f ( G ) ≤ Δ + 3 if Δ ≥ 4 , and χ e f ( G ) ≤ 7 if Δ ≤ 3 .