摘要
An embedding of a digraph in an orientable surface is an embedding as the underlying graph and arcs in each region force a directed cycle. The directed genus is the minimum genus of surfaces in which the digraph can be directed embedded. Bonnington, Conder, Morton and McKenna [J. Cornbin. Theory Ser. B,85（2002） 1-20] gave the problem that which tournaments on n vertices have the directed genus [（n-3）（n-4）/12]the genus of Kn. In this paper, we use the current graph method to show that there exists a tournament, whichhas the directed genus[（n-3）（n-4）/12], on n vertices if and only if n = 3 or 7 （mod 12）.
An embedding of a digraph in an orientable surface is an embedding as the underlying graph and arcs in each region force a directed cycle. The directed genus is the minimum genus of surfaces in which the digraph can be directed embedded. Bonnington, Conder, Morton and McKenna [J. Cornbin. Theory Ser. B,85（2002） 1-20] gave the problem that which tournaments on n vertices have the directed genus [（n-3）（n-4）/12]the genus of Kn. In this paper, we use the current graph method to show that there exists a tournament, whichhas the directed genus[（n-3）（n-4）/12], on n vertices if and only if n = 3 or 7 （mod 12）.
基金
Supported by the National Natural Science Foundation of China(No.11731002)
the Fundamental Research Funds for the Central Universities(Nos.2016JBM071,2016JBZ012)
