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 @D. The edge-face chromatic number @g"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 @g"e"f(G)@[email protected]+1 if @D>=13, @g"e"f(G)@[email protected]+2 if @D>=12, @g"e"f(G)@[email protected]+3 if @D>=4, and @g"e"f(G)@?7 if @[email protected]?3.

[1]  Bojan Mohar,et al.  Graphs of degree 4 are 5-edge-choosable , 1999 .

[2]  Ko-Wei Lih,et al.  A new proof of Melnikov's conjecture on the edge-face coloring of plane graphs , 2002, Discret. Math..

[3]  Yue Zhao,et al.  Coloring edges of graphs embedded in a surface of characteristic zero , 2003, J. Comb. Theory, Ser. B.

[4]  K. Appel,et al.  Every Planar Map Is Four Colorable , 2019, Mathematical Solitaires & Games.

[5]  Alexandr V. Kostochka,et al.  List Edge and List Total Colourings of Multigraphs , 1997, J. Comb. Theory B.

[6]  Limin Zhang,et al.  Every Planar Graph with Maximum Degree 7 Is of Class 1 , 2000, Graphs Comb..

[7]  Yue Zhao,et al.  On Improving the Edge-Face Coloring Theorem , 2001, Graphs Comb..

[8]  Oleg V. Borodin.  Simultaneous coloring of edges and faces of plane graphs , 1994, Discret. Math..

[9]  Daniel P. Sanders,et al.  On simultaneous colorings of embedded graphs , 2000, Discret. Math..

[10]  G. Ringel,et al.  Solution of the heawood map-coloring problem. , 1968, Proceedings of the National Academy of Sciences of the United States of America.

[11]  Ko-Wei Lih,et al.  The edge-face choosability of plane graphs , 2004, Eur. J. Comb..

[12]  Yue Zhao,et al.  On Simultaneous Edge-Face Colorings of Plane Graphs , 1997, Comb..

[13]  Yue Zhao,et al.  Planar Graphs of Maximum Degree Seven are Class I , 2001, J. Comb. Theory B.

[14]  Adrian O. Waller.  Simultaneously Colouring the Edges and Faces of Plane Graphs , 1997, J. Comb. Theory, Ser. B.

[15]  G. Ringel.  Map Color Theorem , 1974 .