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_ef(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_ef(G)≤Δ+1 if Δ≥13, χ_ef(G)≤Δ+2 if Δ≥12, χ_ef(G)≤Δ+3 if Δ≥4, and χ_ef(G)≤7 if Δ≤3.