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

Let GG be a graph embedded in a surface of characteristic zero with maximum degree ΔΔ. The edge-face chromatic number χef(G)χef(G) of GG 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 χef(G)≤Δ+1χef(G)≤Δ+1 if Δ≥13Δ≥13, χef(G)≤Δ+2χef(G)≤Δ+2 if Δ≥12Δ≥12, χef(G)≤Δ+3χef(G)≤Δ+3 if Δ≥4Δ≥4, and χef(G)≤7χef(G)≤7 if Δ≤3Δ≤3.