In this paper we enumerate the rooted dual loopless nonseparable near-triangular maps on the sphere and the projective plane with the valency of root-face and the number of inner faces as parameters. Explicit expressions of enumerating functions are derived for such maps on the sphere and the projective plane. A parametric expression of the generating function is obtained for the rooted 2-connected triangular maps on the projective plane, from which asymptotics evaluations are derived.
Cycle base theory of a graph has been well studied in abstract mathematical field such matroid theory as Whitney and Tutte did and found many applications in prat-ical uses such as electric circuit theory and structure analysis, etc. In this paper graph embedding theory is used to investigate cycle base structures of a 2-(edge)-connected graph on the sphere and the projective plane and it is shown that short cycles do generate the cycle spaces in the case of 'small face-embeddings'. As applications the authors find the exact formulae for the minimum lengthes of cycle bases of some types of graphs and present several known results. Infinite examples shows that the conditions in their main results are best possible and there are many 3-connected planar graphs whose minimum cycle bases can not be determined by the planar formulae but may be located by re-embedding them into the projective plane.
结合边连通度,探讨了独立集中具有最小特定度和的点的上可嵌入图.得到了下列结果.(1)设G是一个2-边连通简单图且满足条件：对任意一个G的3-独立集I,x_i,x_j∈I(i,j=1,2,3),d(x_i,x_j)≥3(1≤i≠j≤3)sum from i=1 to 3 d(x_i)≥v+1 (v=V(G)),则G是上可嵌入的；(2)设G是一个3-边连通简单图且满足条件：对任意一个G的6-独立集I,x_i,x_j∈I(i,j=1,2,3,4,5,6),d(x_i,x_j)≥3(1≤i≠j≤6)sum from i=1 to 6 d(x_i)≥v+1(v=|V(G)|),则G是上可嵌入的.
In this paper, we show that for a locally LEW-embedded 3-connected graph G in orientable surface, the following results hold： 1） Each of such embeddings is minimum genus embedding; 2） The facial cycles are precisely the induced nonseparating cycles which implies the uniqueness of such embeddings; 3） Every overlap graph O（G, C） is a bipartite graph and G has only one C-bridge H such that C U H is nonplanar provided C is a contractible cycle shorter than every noncontractible cycle containing an edge of C. This extends the results of C Thomassen＇s work on LEW-embedded graphs.
